Maths WorkshopMaths problems with full solutions

Maths problems with solutions

2245 problems · A growing collection of original maths problems with hints and complete solutions: algebra, analysis, linear algebra, probability, number theory and more.

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Ex.1A functional equation with f(2x)

#functional equations #continuity at a point #sequences

Functional equations

Ex.2A recurrence and a proof by induction

#induction #recurrence #closed form

Ex.3Cauchy's equation f(x + y) = f(x) + f(y)

#functional equations #density #rational numbers

Functional equations

Ex.4Continuity with parameters

#continuity #piecewise functions #parameters

Continuous and discontinuous functions

Ex.5When do integrals of powers converge?

#improper integrals #p-test #Riemann integrals

Improper integrals

Ex.6Extending a rational function by continuity

#extension by continuity #factoring #removable singularity

Continuous and discontinuous functions

Ex.7An integral whose integrand blows up

#improper integrals #logarithm #limits

Improper integrals

Ex.8Polynomial bounds for sine and cosine

#inequalities #monotonicity

Differentiable functions

Ex.9Terms that do not tend to zero

#divergence test #necessary condition

Convergence tests

Ex.10Expanding √x at x = 4

#Taylor expansion #shifted point #numerical estimate

Taylor expansions

Ex.11Product of irrational and nonzero rational

#irrationality #product #rational

Ex.12Sum of irrational and rational

#irrationality #sum #rational

Ex.13Checking linearity

#linear map #definition #examples

Ex.14Matrix addition and scalar multiplication

#matrix operations #addition #scalar multiplication

Matrices and linear systems

Ex.15A fence along a river

#optimisation #area #quadratic function

Optimisation and curve sketching

Ex.16f(x)f(y) − f(xy) = x + y

#substitution #special values #verification

Functional equations

Ex.17The mean value theorem

#mean value theorem #Lagrange #existence

Mean Value Theorem

Ex.18Rolle's theorem on a polynomial

#Rolle #polynomial #existence

Mean Value Theorem

Ex.19Step functions and their integrals

#step function #partition #integral

Riemann integral

Ex.20Integral of a constant step function

#constant #step function #elementary

Riemann integral

Ex.21An improper integral with exponentials

#improper integral #exponential #partial fraction

Improper integrals

Ex.22Three basic limits by conjugation and standard limits

#conjugate #standard limit #sine

Ex.23Irrationality of √2 + √3

#irrational #proof by contradiction

Ex.24Equivalents, and a sign trap

#equivalents #comparison test #positive series

Convergence tests

Ex.25A fixed point for every continuous self-map of [0,1]

#fixed points #intermediate value theorem

Intermediate Value Theorem

Ex.26Extensions by continuity and oscillation

#extension by continuity #oscillation #squeeze theorem

Continuous and discontinuous functions

Ex.27Locating the roots of a cubic

#intermediate value theorem #polynomial roots #bracketing

Intermediate Value Theorem

Ex.28How long is the drug effective?

#applications #intermediate value theorem #bracketing

Intermediate Value Theorem

Ex.29An improper integral of a rational function

#improper integrals #arctangent #convergence

Improper integrals

Ex.30Where is 1/ln|x| continuous?

#continuity #limits #logarithm

Continuous and discontinuous functions

Ex.31Which of these functions extend by continuity?

#extension by continuity #one-sided limits #exponential

Continuous and discontinuous functions

Ex.32lim (1/x² − 1/sin² x)

#limits #expansions #difference of fractions

Taylor expansions

Ex.33lim ((1 + x)^(1/x) / e)^(1/x)

#limits #indeterminate form 1^∞ #logarithm #expansions

Taylor expansions

Ex.34Summary of continuity

#summary #continuity

Ex.35Eigenspaces are subspaces

#eigenspace #subspace #linear map

Eigenvalues and diagonalisation

Ex.36Irrationality of ∛2

#cube root #irrationality #unique factorization

Ex.37Between any two rationals lies an irrational

#density #irrationals #betweenness

Ex.38Irrationality of √p for prime p

#sqrt p #prime #irrationality

Ex.39Irrationality of √2

#sqrt2 #irrationality #proof by contradiction

Ex.40Irrationality of √3

#sqrt3 #irrationality #mod 3

Ex.41Linear independence of logs

#logarithm #independence #irrationality

Ex.42lim cos(1/n)^(n²)

#sequences #expansions #indeterminate form 1^∞

Taylor expansions

Ex.43Mean value theorem for $f(x)=x^3$

#mean value theorem #cubic #existence

Mean Value Theorem

Ex.44Summary of the mean value theorem

#summary #MVT #Rolle

Mean Value Theorem

Ex.45Additivity over adjacent intervals

#additivity #step function #split

Riemann integral

Ex.46A step function built from the floor

#step function #floor function #integral

Riemann integral

Ex.47Linearity of the integral on step functions

#linearity #step functions #sum

Riemann integral

Ex.48Monotonicity of the integral

#monotonicity #step functions #comparison

Riemann integral

Ex.49Summary of step functions and Riemann integration

#summary #step functions #Riemann integrability

Riemann integral

Ex.50Two subdivisions for the same function

#step function #subdivision #independence

Riemann integral

Ex.51Four limits with absolute value and trigonometry

#absolute value #one-sided limits #factorization

Ex.52Three more classical limits

#squeeze theorem #logarithm #exponential form

Ex.53Proving convergence by comparison

#improper integrals #comparison test #divergence

Improper integrals

Ex.54Floor plus square root of the fractional part

#continuity #floor function #one-sided limits

Continuous and discontinuous functions

Ex.55The monk on the mountain

#intermediate value theorem #classic puzzle

Intermediate Value Theorem

Ex.56Expansions of 1/(1 + x + x²) and ln(1 + x + x²)

#Taylor expansion #geometric series #factorisation trick

Taylor expansions

Ex.57Irrationality of ⁿ√2

#nth root #irrationality #prime factorization

Ex.58Irrationality of log₁₀ 2

#logarithm #irrationality #base 10

Ex.59Irrationality of log₂ 3

#logarithm #irrationality #base change

Ex.60A square root in disguise

#irrationality #quadratic #conjugates

Ex.61Ratios involving irrationals

#ratio #irrationality #algebraic

Ex.62Position of a curve relative to its tangent

#tangent line #inflection #local behaviour

Taylor expansions

Ex.63Integral of $|f|$ for a step function

#absolute value #step function #triangle inequality

Riemann integral

Ex.64Darboux sums for a step function

#Darboux sums #upper lower #step functions

Riemann integral

Ex.65A step function with zero integral

#zero integral #step functions #nonnegative

Riemann integral

Ex.66Squeezing between step functions

#Riemann integrability #step functions #squeeze

Riemann integral

Ex.67A convergent improper integral of $\ln x/(1+x)^2$

#improper integral #integration by parts #logarithm

Improper integrals

Ex.68An improper integral of $\ln x/x^2$ by parts

#improper integral #integration by parts #logarithm

Improper integrals

Ex.69Three limits involving logarithm and exponentials

#exponential form #logarithm #indeterminate forms

Ex.70The midpoint functional equation

#functional equations #affine functions

Functional equations

Ex.71Decimal expansions and rationality

#decimal expansion #rational #periodic

Ex.72Irrationality of a cubic root expression

#cubic root #irrationality #algebraic

Ex.73Irrationality of √2 + √3

#algebraic #sum of roots #irrationality

Ex.74A sum of radicals

#radicals #irrationality #algebraicity

Ex.75Composition with a continuous function

#composition #continuous #integrability preservation

Riemann integral

Ex.76A monotone function on a fine grid

#monotone functions #Riemann integrability #step approximation

Riemann integral

Ex.77Continuity of a rational function

#rational function #domain #continuity

Ex.78Glued along the rationals and the irrationals

#continuity #density #sequences #pathological functions

Continuous and discontinuous functions

Ex.79The limit of an iterated sequence is a fixed point

#sequences #fixed points #continuity #golden ratio

Continuous and discontinuous functions

Ex.80Maximum and minimum of two continuous functions

#continuity #absolute value #algebra of continuous functions

Continuous and discontinuous functions

Ex.81The absolute value of a continuous function

#continuity #absolute value #counterexamples

Continuous and discontinuous functions

Ex.82The ε–N definition of a limit

#definition of a limit #epsilon-N #divergence

Ex.83How many terms are needed to compute sin 1?

#Taylor-Lagrange #error bound #numerical computation

Taylor expansions

Ex.84Rational root theorem and irrationality

#rational root theorem #irrationality #cubic

Ex.85Matrix of a linear map

#matrix representation #canonical basis #linear map

Ex.86How many trees in the orchard?

#optimisation #modelling #quadratic function

Optimisation and curve sketching

Ex.87Fixed point on [0,1]

#fixed point #IVT #auxiliary function

Intermediate Value Theorem

Ex.88A continuous periodic function is bounded

#extreme value theorem #periodic functions #boundedness

Continuous and discontinuous functions

Ex.89Sum of irrationals in disguise

#irrationality #sum #structure

Ex.90Irrationality of π

#pi #irrationality #Lambert

Ex.91Continued fractions and irrationality

#continued fractions #irrationality #sqrt2

Ex.92Irrationality of √n for non-square n

#sqrt n #non-square #irrationality

Ex.93Additive and multiplicative functions

#Cauchy equation #squares are nonnegative #monotonicity

Functional equations

Ex.94Cauchy's equation with monotonicity

#Cauchy equation #rationals #density

Functional equations

Ex.95x³ + 2y³ + 4z³ = 0

#infinite descent #parity #irrationality

Ex.96A parameter integral equal to arctan

#parameter integral #fundamental theorem #arctan

Riemann integral

Ex.97A limit via the mean value theorem for integrals

#mean value theorem for integrals #logarithm #limit

Riemann integral

Ex.98Liouville numbers

#Liouville #transcendental #irrationality

Ex.99A general irrationality criterion

#criterion #irrationality #polynomial

Ex.100f(x + y) + f(x − y) = 2f(x)f(y)

#d'Alembert #cosine #differential equation #differentiability

Functional equations

Ex.101f(x² + f(y)) = y + f(x)²

#injectivity #surjectivity #oddness #Cauchy equation

Functional equations

Ex.102When the equivalence rule for improper integrals fails

#improper integral #integration by parts #absolute convergence #conditional convergence

Improper integrals

Ex.103Limit of an integral with arctan

#mean value theorem for integrals #arctan #limit

Riemann integral

Ex.104Limit of an integral with cosine and logarithm

#mean value theorem for integrals #cosine #logarithm

Riemann integral

Ex.105Commuting maps on $[0,1]$ share a fixed point

#commuting maps #IVT #extreme value theorem #induction

Intermediate Value Theorem

Ex.106Lipschitz functions are uniformly continuous

#Lipschitz #uniform continuity #epsilon-delta

Continuous and discontinuous functions

Ex.107Coordinate projections are continuous

#projection #continuity #coordinates

Ex.108Computing a kernel

#kernel #linear map #basis

Ex.109Distance to a line, without a formula

#optimisation #distance #line #quadratic function

Optimisation and curve sketching

Ex.110Continuous functions with f^2=1 on an interval

#IVT #sign #constant functions

Intermediate Value Theorem

Ex.1111/x on (0,1] is not uniformly continuous

#uniform continuity #open intervals #Heine theorem

Continuous and discontinuous functions

Ex.112The square root: uniformly continuous, not Lipschitz

#uniform continuity #Lipschitz #square root

Continuous and discontinuous functions

Ex.113x² is not uniformly continuous on the real line

#uniform continuity #sequences #counterexamples

Continuous and discontinuous functions

Ex.114A continuous f with f o f= cos cannot exist

#functional equation #cosine #continuity #monotonicity

Functional equations

Ex.115Gelfond–Schneider theorem (statement)

#Gelfond-Schneider #transcendence #Hilbert 7th

Ex.116Least squares for a line through the origin

#least squares #regression #statistics #quadratic

Optimisation and curve sketching

Ex.117Limits of sums as Riemann sums

#Riemann sums #limits #pi #ln 2

Riemann integral

Ex.118Basis of a kernel

#kernel #basis #linear map

Ex.119A limit of a product via Riemann sums

#Riemann sums #logarithm #limit #product

Riemann integral

Ex.120Computing an image

#image #linear map #generators

Ex.121The most economical speed of a truck

#optimisation #modelling #economics #cost per kilometre

Optimisation and curve sketching

Ex.122Fractional parts of multiples of √2

#fractional parts #density #irrational

Ex.123The economic order quantity

#optimisation #economics #inventory #modelling

Optimisation and curve sketching

Ex.124Rank of a linear map

#rank #image dimension #matrix

Ex.125Maximising profit

#optimisation #economics #marginal cost #modelling

Optimisation and curve sketching

Ex.126Rank-nullity theorem on an example

#rank-nullity #rank #kernel dimension

Ex.127The strongest beam cut from a log

#optimisation #geometry #engineering #Pythagoras

Optimisation and curve sketching

Ex.128An injective linear map

#injectivity #kernel #linear map

Ex.129A window with a semicircular top

#optimisation #area #perimeter #geometry

Optimisation and curve sketching

Ex.130A surjective linear map

#surjectivity #rank #linear map

Ex.131An isomorphism

#isomorphism #inverse #linear map

Ex.132The trapezoid of largest area with three equal sides

#optimisation #trigonometry #geometry #trapezoid

Optimisation and curve sketching

Ex.133Similar matrices have the same eigenvalues

#similarity #eigenvalues #characteristic polynomial

Eigenvalues and diagonalisation

Ex.134Inverse of an endomorphism

#inverse #endomorphism #matrix inverse

Ex.135The triangle of least area through a point

#optimisation #lines #triangle #area

Optimisation and curve sketching

Ex.136Composition and matrix product

#composition #matrix product #representation

Ex.137Among triangles with a given base and perimeter, the isosceles one is the largest

#optimisation #Heron's formula #triangle

Optimisation and curve sketching

Ex.138Composition of linear maps

#composition #matrix product #linear maps

Ex.139The largest rectangle inscribed in an ellipse

#optimisation #ellipse #parametrisation #trigonometry

Optimisation and curve sketching

Ex.140Kernel of a composition

#composition #kernel #inclusion

Ex.141The largest cone and the largest cylinder inscribed in a sphere

#optimisation #volume #sphere #solids

Optimisation and curve sketching

Ex.142The paper cup of largest capacity

#optimisation #cone #volume #geometry

Optimisation and curve sketching

Ex.143Image of a composition

#composition #image #inclusion

Ex.144At which height should a lamp be hung?

#optimisation #physics #illumination #Lambert's law

Optimisation and curve sketching

Ex.145Rank inequality for a composition

#rank #composition #inequality

Ex.146Rank is invariant under invertible multiplications

#rank #invertible matrices #invariance

Ex.147Maximum sustainable yield

#optimisation #modelling #ecology #logistic growth

Optimisation and curve sketching

Ex.148A matrix with a single eigenvalue

#eigenvalues #eigenspace #diagonalization #Jordan block

Eigenvalues and diagonalisation

Ex.149The shortest path touching a line

#optimisation #reflection #geometry #sums of square roots

Optimisation and curve sketching

Ex.150Change of basis for a linear map

#change of basis #matrix representation #similarity

Ex.151Invariants under similarity

#similarity #trace #determinant #rank #characteristic polynomial

Ex.152The closest point of an ellipse

#optimisation #ellipse #distance #constrained minimum

Optimisation and curve sketching

Ex.153A linear projection

#projection #idempotence #kernel and image

Ex.154Minimising a/x + b/(1 − x)

#optimisation #inequalities #Cauchy–Schwarz #parameters

Optimisation and curve sketching

Ex.155A linear symmetry

#symmetry #involution #linear map

Ex.156The best constant approximating x²

#minimax #uniform approximation #absolute value

Optimisation and curve sketching

Ex.157A nilpotent endomorphism

#nilpotent #kernel #image

Ex.158Firing up a slope: the best angle

#optimisation #trigonometry #physics #projectile

Optimisation and curve sketching

Ex.159The maximum of 3 sin x + 4 cos x

#extrema #trigonometry #amplitude

Optimisation and curve sketching

Ex.160Kernel and image of an endomorphism - when are they supplementary?

#kernel #image #direct sum #endomorphism

Ex.161Global extrema of x³ − 3x on [−2, 3]

#global extrema #closed interval #critical points

Optimisation and curve sketching

Ex.162Projection onto a subspace

#orthogonal projection #linear map #kernel and image

Ex.163Extrema of |x² − 3x + 2| on [0, 3]

#absolute value #global extrema #non-differentiable points

Optimisation and curve sketching

Ex.164Trigonalization and eigenvalues

#trigonalization #characteristic polynomial #real eigenvalues

Eigenvalues and diagonalisation

Ex.165Kernel and image of a matrix and its transposition

#kernel #image #transpose #orthogonal complement

Ex.166Eigenvectors and injectivity

#eigenvectors #injectivity #kernel

Ex.167The maximum of xⁿe^(−x)

#extrema #inequalities #exponential #growth

Optimisation and curve sketching

Ex.168The minimum of x^x

#logarithmic differentiation #extrema #limits

Optimisation and curve sketching

Ex.169Diagonalizability and the sum of eigenspaces

#diagonalization #eigenspaces #dimension

Ex.170Invariant subspaces

#invariant subspace #endomorphism #restriction

Ex.171A family of functions: x² + m/x

#parameters #extrema #monotonicity #discussion

Optimisation and curve sketching

Ex.172Complete study of x e^(−x)

#curve sketching #inflection point #number of solutions

Optimisation and curve sketching

Ex.173Induced map on a quotient

#quotient space #induced map #well-defined

Ex.174Functions of a diagonalizable matrix

#functions of matrices #diagonalization #polynomial #well-definedness

Eigenvalues and diagonalisation

Ex.175Complete study of (ln x)/x

#curve sketching #asymptotes #inflection point #logarithm

Optimisation and curve sketching

Ex.176Sketching x³/(x² − 1)

#curve sketching #oblique asymptote #odd function #inflection point

Optimisation and curve sketching

Ex.177Sketching x√(1 − x²)

#curve sketching #vertical tangent #odd function #domain

Optimisation and curve sketching

Ex.178How many solutions does eˣ = ax have?

#number of solutions #parameter #tangent #extrema

Optimisation and curve sketching

Ex.179A classic inequality

#inequalities #AM-GM

Ex.180A nested radical

#radicals #identities

Equations and radicals

Ex.181Area from coordinates

Analytic geometry

Ex.182Congruence is an equivalence relation

#relations #modular arithmetic

Ex.183De Morgan's law for sets

Ex.184The exponential lies above its tangent

#inequalities #derivatives

Differentiation

Ex.185Four-digit numbers with distinct digits

#counting #arrangements

Counting principles

Ex.186Injective or surjective?

#maps #bijection

Ex.187Integral of sine squared

#integrals #trigonometry

Integration techniques

Ex.188Integration by parts

#integrals #by parts

Integration techniques

Ex.189Inverse of a 2×2 matrix

#inverse #determinants

Matrices and linear systems

Ex.190A power of 1 + i

#complex numbers

Complex numbers

Ex.191Squares and pairwise products

#inequalities #squares

Ex.192(e^(2x) − 1) over x

#exponential #change of variable

Ex.193A limit by factoring

#indeterminate forms #factoring

Ex.194Limits of rational functions at infinity

#limits at infinity #rational functions

Ex.195One-sided limits of |x|/x

#one-sided limits #absolute value

Ex.196sin(3x) over 5x

#fundamental trigonometric limit #change of variable

Ex.197Sine over x at infinity

#squeeze theorem #limits at infinity

Ex.198A telescoping sum

#telescoping #series #partial sums

Ex.199Area between a parabola and a line

#area #definite integrals #intersection points

Riemann integral

Ex.200An area with an absolute value

#definite integrals #absolute value #sign of a function

Riemann integral

Ex.201Arithmetic and geometric sequences

#arithmetic sequence #geometric sequence #sums

Ex.202Recognising antiderivatives

#antiderivatives #definite integrals #fundamental theorem

Riemann integral

Ex.203Derivative of a quotient

#derivative #quotient rule

Differentiable functions

Ex.204Basic divisibility and linear combinations

#divisibility #linear combinations #divisors

Divisibility and gcd

Ex.205Consecutive integers are coprime

#coprime #consecutive integers #Euclidean algorithm

Divisibility and gcd

Ex.206Finding divisors systematically

#divisors #enumeration #bounds

Divisibility and gcd

Ex.207Geometric Series

#geometric series #convergence #summation

Ex.208Geometric series and repeating decimals

#geometric series #decimals #sum

Ex.209Geometric series with a parameter

#geometric series #convergence domain #parameters

Ex.210Integrals that simplify by parity

#symmetry #odd and even functions

Riemann integral

Ex.211Rolle's theorem on a quartic

#Rolle's theorem #polynomials

Mean Value Theorem

Ex.212The sum of the first n squares

#induction #sums #limits

Ex.213Derivative and tangent line of a cubic

#derivative #tangent line #polynomials

Differentiable functions

Ex.214The derivative from the definition

#definition of the derivative #difference quotient #square root

Differentiable functions

Ex.215Definition and basic values

#binomial coefficient #definition #factorial

Binomial coefficients

Ex.216Pascal's triangle

#Pascal triangle #rows #computation

Binomial coefficients

Ex.217Symmetry of binomial coefficients

#symmetry #binomial coefficient #identity

Binomial coefficients

Ex.218Cardinality of finite sets

#finite sets #cardinality #bijection

Countability and cardinality

Ex.219$\mathbb{N}$ is countably infinite

#naturals #countably infinite #identity

Countability and cardinality

Ex.220Basic operations on complex numbers

#complex numbers #addition #multiplication #conjugate

Complex numbers

Ex.221The equation $z=\bar z$

#conjugate #real axis #equation

Complex numbers

Ex.222Modulus via the conjugate

#modulus #conjugate #formula

Complex numbers

Ex.223Arithmetic modulo $n$

#modular arithmetic #operations #compatibility

Ex.224Congruence as an equivalence relation

#equivalence relation #congruence #residue classes

Ex.225Continuity of a polynomial

#continuity #polynomial #continuity everywhere

Ex.226Counting functions

#functions #product rule #counting

Counting principles

Ex.227Handshakes at a party

#handshake #edges #complete graph

Counting principles

Ex.228The pigeonhole principle

#pigeonhole #Dirichlet #basic

Counting principles

Ex.229The product rule

#product rule #multiplication principle #independent choices

Counting principles

Ex.230The sum rule

#sum rule #addition principle #disjoint cases

Counting principles

Ex.231Determinant of a 2×2 matrix

#determinant #2x2 matrices #invertibility

Ex.232Divisibility rules

#divisibility rules #modular arithmetic #base 10

Divisibility and gcd

Ex.233GCD of a GCD

#gcd #associativity #three numbers

Divisibility and gcd

Ex.234The HCF of consecutive integers

#coprime #consecutive integers #gcd

Divisibility and gcd

Ex.235A linear equation

#linear equation #solving #basic

Equations and radicals

Ex.236A quadratic equation

#quadratic equation #discriminant #roots

Equations and radicals

Ex.237Basic graph terminology

#graph #vertices #edges #degree

Ex.238The complete graph $K_n$

#complete graph #edges #degree

Ex.239Parity of odd-degree vertices

#parity #degree #handshake

Ex.240The handshake lemma

#handshake lemma #degree sum #parity

Ex.241Additive and multiplicative notations

#notation #abelian group #convention

Ex.242Checking the group axioms

#group axioms #examples #counterexamples

Ex.243The two-set inclusion-exclusion formula

#inclusion-exclusion #two sets #union

Inclusion-exclusion

Ex.244Disjoint unions

#IE #disjoint #sum

Inclusion-exclusion

Ex.245Proof of the two-set formula

#IE #two sets #proof

Inclusion-exclusion

Ex.246Inner product axioms

#inner product #axioms #dot product

Inner product spaces

Ex.247Sum of irrationals can be rational

#irrationality #sum #counterexample

Ex.248Proof by counterexample

#counterexample #disproof #universal statement

Proof techniques

Ex.249De Morgan's laws

#De Morgan #logic #negation

Proof techniques

Ex.250Direct proof

#direct proof #implication #even odd

Proof techniques

Ex.251Geometric sum by induction

#induction #geometric series #sum

Ex.252Proof by induction: sum of first $n$ integers

#induction #sum #integers

Ex.253Summary of induction

#summary #induction #techniques

Ex.254Metric spaces

#metric #definition #axioms

Metric and normed spaces

Ex.255Open balls in a metric space

#open ball #metric #definition

Metric and normed spaces

Ex.256Double integral over a rectangle

#double integral #rectangle #iterated

Multiple integrals

Ex.257Evaluating partial derivatives at a point

#partial derivatives #evaluation #point

Partial derivatives

Ex.258First partial derivatives

#partial derivatives #definition #computation

Partial derivatives

Ex.259The pigeonhole principle on birthdays

#pigeonhole #birthday #basic application

Pigeonhole principle

Ex.260Socks in the dark

#pigeonhole #socks #drawers

Pigeonhole principle

Ex.261Degree of sums and products

#degree #operations #polynomial

Ex.262GCD of successive integers

#coprime #consecutive integers #gcd

Ex.263Distinct primes are coprime

#coprime #primes #distinct

Ex.264Definition and first examples of primes

#primes #definition #examples

Ex.265GCD with a prime

#gcd #prime #divisor

Ex.266Definition and matrix of a quadratic form

#quadratic form #symmetric matrix #definition

Quadratic forms

Ex.267The existential quantifier

#existential quantifier #exists #example

Quantifiers and negation

Ex.268The universal quantifier

#universal quantifier #forall #proposition

Quantifiers and negation

Ex.269Double negation

#double negation #logic #equivalence

Quantifiers and negation

Ex.270Proving an existential statement

#existential proof #example #witness

Quantifiers and negation

Ex.271Universal over a finite set

#finite quantifier #forall #conjunction

Quantifiers and negation

Ex.272Binary relations

#binary relation #definition #examples

Ex.273Reflexive relations

#reflexive #relation #definition

Ex.274Symmetric relations

#symmetric #relation #definition

Ex.275Transitive relations

#transitive #relation #definition

Ex.276Checking the ring axioms

#ring axioms #examples #counterexamples

Rings and fields

Ex.277Basic set operations

#sets #union #intersection #complement

Ex.278The empty set

#empty set #uniqueness #axioms

Ex.279Subset relations

#subset #inclusion #definition

Ex.280Open balls in $\mathbb{R}$

#open ball #interval #topology

Topology of ℝ

Ex.281Open intervals are open sets

#open sets #open intervals #definition

Topology of ℝ

Ex.282Closed sets in $\mathbb{R}^n$

#closed sets #complement #examples

Topology of ℝⁿ

Ex.283Open balls in $\mathbb{R}^n$

#open ball #Euclidean norm #definition

Topology of ℝⁿ

Ex.284Open sets in $\mathbb{R}^n$

#open sets #definition #ball

Topology of ℝⁿ

Ex.285Checking the vector space axioms

#vector space axioms #examples #counterexamples

Vector spaces and subspaces

Ex.286Cartesian product with seven elements

#Cartesian product #cardinality #impossible

Ex.287Odd-sized real matrices have a real eigenvalue

#intermediate value theorem #characteristic polynomial #eigenvalue

Applications and the case n = 2

Ex.288The inverse does not extend continuously across singular matrices

#inversion #unboundedness #continuity #extension

Applications and the case n = 2

Ex.289The closed set det(A - I) = 0 has empty interior

#hypersurface #determinant #closed set

Density and closures

Ex.290Convergence of the matrix exponential and its first bounds

#matrix exponential #absolute convergence #submultiplicative norm

Exponential, logarithm and group topology

Ex.291The exponential of a sum: commuting and non-commuting matrices

#matrix exponential #commutation #counterexample

Exponential, logarithm and group topology

Ex.292Every norm sphere of M_n(R) is a sphere

#norms #sphere #homeomorphism #radial projection

Rank, spectrum and norms

Ex.293Trace and determinant bounded by the operator norm

#operator norm #singular values #trace #determinant

Rank, spectrum and norms

Ex.294The Frobenius norm comes from an inner product

#Frobenius norm #inner product #trace

Norms and metric structure

Ex.295Two norms of the identity matrix

#Frobenius norm #operator norm #identity matrix

Norms and metric structure

Ex.296Entries versus Frobenius norm

#Frobenius norm #max norm #equivalence of norms

Norms and metric structure

Ex.297Every norm on M_n(R) is complete

#completeness #finite dimension #equivalence of norms

Norms and metric structure

Ex.298Column-sum and row-sum norms of a 2×2 matrix

#matrix norms #column sum #row sum #computation

Norms and metric structure

Ex.299The groups GL_n and SL_n: open and closed

#open sets #closed sets #determinant #GL_n #SL_n

Open, closed, interior, boundary

Ex.300Proper subspaces are closed with empty interior

#subspaces #interior #closed sets #finite dimension

Open, closed, interior, boundary

Ex.301Closed or open? Three classical subspaces

#subspaces #symmetric matrices #trace #triangular matrices

Open, closed, interior, boundary

Ex.302Integer matrices are closed and discrete, rational ones dense

#integer matrices #discrete sets #density #rationals

Open, closed, interior, boundary

Ex.303Polynomial maps between matrix spaces are continuous

#polynomial maps #continuity #determinant #adjugate

Continuity of matrix maps

Ex.304Matrix multiplication is jointly continuous

#multiplication #bilinear map #continuity #submultiplicativity

Continuity of matrix maps

Ex.305Inversion is a homeomorphism of GL_n

#inverse #Cramer's rule #homeomorphism #adjugate

Continuity of matrix maps

Ex.306Which natural functions of a matrix are continuous?

#continuity #rank #kernel #eigenvalues #counterexamples

Continuity of matrix maps

Ex.307GL_n is a topological group

#topological group #determinant #homomorphism #SL_n

Continuity of matrix maps

Ex.308Components of the linear group

#connected components #general linear group #determinant #local connectedness

Ex.309Convex sets of matrices are path connected

#connectedness #convex sets #path connected #matrices

Ex.310Singular matrices and the nilpotent cone are star-shaped

#connectedness #singular matrices #nilpotent matrices #star-shaped

Ex.311The special linear group is totally disconnected

#totally disconnected #discrete subspace #integer matrices

Ex.312SO(2) is a circle, O(2) has two components

#SO(2) #O(2) #rotations #reflections #path components

Ex.313Components of the set where the trace is non-zero

#connected components #trace #half-spaces

Ex.314A limit with sine

#limits #taylor

Ex.315Sum of binomial coefficients

#binomial #counting

Binomial coefficients

Ex.316A cubic with exactly one real root

#intermediate value theorem #monotonicity

Intermediate Value Theorem

Ex.317A large power modulo 7

#congruences #periodicity

Ex.318A limit with the exponential

#limits #taylor

Ex.319A plane as a subspace

#subspaces #basis

Vector spaces and subspaces

Ex.320An equation with absolute values

#absolute value #equations

Equations and radicals

Ex.321An improper integral

#improper integrals #integration by parts

Improper integrals

Ex.322Arrangements of MISSISSIPPI

#counting #arrangements

Counting principles

Ex.323Bernoulli's inequality

#induction #inequalities

Ex.324Distance from a point to a line

#lines #projection

Analytic geometry

Ex.325The Euclidean algorithm and Bézout

Divisibility and gcd

Ex.326There are infinitely many primes

#primes #proof by contradiction

Ex.327Inverting I − A

#matrices #nilpotent

Matrices and linear systems

Ex.328Kernel of a matrix

Matrices and linear systems

Ex.329Negating convergence

#quantifiers #sequences

Quantifiers and negation

Ex.330Pascal's rule

#binomial #counting

Binomial coefficients

Ex.331Pigeonhole modulo 4

#pigeonhole #congruences

Pigeonhole principle

Ex.332Radius of convergence

#power series #ratio test

Ex.333Remainder of a polynomial

#polynomials #euclidean division

Ex.3346 divides n³ − n

#divisibility #factorisation

Divisibility and gcd

Ex.335Stars and bars

#counting #equations

Counting principles

Ex.336Sum of cubes of the roots

#vieta #polynomials

Ex.337Sum of squares by induction

#induction #sums

Ex.338Sums of two squares modulo 4

#congruences #squares

Ex.339Thales' theorem

#circles #angles

Ex.340The handshake lemma

#graphs #parity

Ex.341The integers are countable

#bijection #countability

Countability and cardinality

Ex.342The last digit of 7^2026

#congruences #periodicity

Ex.343The logarithm lies below its tangent

#inequalities #derivatives

Differentiation

Ex.344The conjugate trick at infinity

#conjugate #square roots #infinity minus infinity

Ex.345The conjugate trick at zero

#conjugate #square roots #indeterminate forms

Ex.346Epsilon-delta for a linear function

#epsilon-delta #definition of limit

Ex.347(e^x − cos x) over sin x

#combining known limits #indeterminate forms

Ex.348One minus cosine over x squared

#trigonometric limits #half-angle formula

Ex.349Sine is 1-Lipschitz

#mean value theorem #inequalities

Mean Value Theorem

Ex.350The n-th root of n

#sequences #logarithm #growth comparison

Ex.351(2^x − 1) over x

#exponential #logarithm #derivative

Ex.352x ln(1 + 1/x) at infinity

#logarithm #number e #limits at infinity

Ex.353x times sin(1/x)

#squeeze theorem #oscillation

Ex.354A continuous function that vanishes on the rationals

#density #continuity #sequences

Continuous and discontinuous functions

Ex.355A linear recurrence of order two

#linear recurrence #characteristic equation

Ex.356A telescoping product

#telescoping #products #factoring

Ex.357A telescoping sum with factorials

#telescoping #factorials

Ex.358A square root whose difference tends to 1/2

#divergence test #square roots #asymptotics

Convergence tests

Ex.359An arithmetico-geometric sequence

#recurrence #fixed point #geometric sequence

Ex.360Odd degree polynomials have a real root

#intermediate value theorem #polynomials #limits at infinity

Intermediate Value Theorem

Ex.361The Bertrand series, all cases

#Bertrand series #parameters #integral test #comparison

Convergence tests

Ex.362Derivative of an inverse function

#inverse function #implicit differentiation #arcsine

Differentiable functions

Ex.363A surprisingly simple derivative

#chain rule #composition #square roots

Differentiable functions

Ex.364Derivative with an absolute value

#differentiability #absolute value #one-sided derivatives

Differentiable functions

Ex.365Differentiability when gluing two pieces

#differentiability #piecewise functions #one-sided derivatives

Differentiable functions

Ex.366Common divisors and common multiples

#divisibility #lcm #structure

Divisibility and gcd

Ex.367Computing LCM from divisibility

#lcm #divisibility #product

Divisibility and gcd

Ex.368$n^2-1$ is divisible by 8 for odd $n$

#divisibility #parity #consecutive integers

Divisibility and gcd

Ex.369The relation between divisibility and lcm

#divisibility #lcm #product relation

Divisibility and gcd

Ex.370Simplifying a fraction with the divisibility

#divisibility #irreducible fraction #simplification

Divisibility and gcd

Ex.371divisibility of three numbers

#divisibility #three numbers #prime factorization

Divisibility and gcd

Ex.372Splitting the sum of 1/n² into even and odd terms

#absolute convergence #rearrangement #pi squared

Ex.373A separable first-order equation

#separable equation #integration #initial condition

First-order equations

Ex.374Integration by parts, two classics

#integration by parts #definite integrals

Integration techniques

Ex.375Three limits by the squeeze theorem

#squeeze theorem #sums #n-th roots

Ex.376Logarithmic differentiation

#derivative #exponential form #powers with variable exponent

Differentiable functions

Ex.377The mean value theorem and an arctangent inequality

#mean value theorem #inequalities #arctangent

Mean Value Theorem

Ex.378Two-sided bounds for the logarithm

#mean value theorem #inequalities #logarithm

Mean Value Theorem

Ex.379Double integral over a triangular domain

#double integral #iterated integration #triangular domain

Multiple integrals

Ex.380Partial derivatives and the gradient

#partial derivatives #gradient #chain rule

Partial derivatives

Ex.381Simplifying and decomposing rational integrands

#partial fractions #rational functions

Integration techniques

Ex.382Complex conjugate roots

#characteristic equation #complex roots #oscillation

Linear equations of order two

Ex.383A double root

#characteristic equation #double root #homogeneous equation

Linear equations of order two

Ex.384Distinct real roots

#characteristic equation #real roots #homogeneous equation

Linear equations of order two

Ex.385Series with factorials and the number e

#exponential series #factorials #e

Ex.386Integration by substitution

#substitution #change of variable

Integration techniques

Ex.387Tangent lines through an external point

#tangent line #parabola #geometry

Differentiable functions

Ex.388Telescoping series with partial fractions

#telescoping #partial fractions #sums

Ex.389Telescoping with logarithms

#telescoping #logarithm #divergence

Ex.390Telescoping with a shift p

#telescoping #partial fractions #harmonic numbers

Ex.391Telescoping with square roots

#telescoping #square roots #divergence #p-series

Convergence tests

Ex.392The bisection method in practice

#bisection #numerical methods #intermediate value theorem

Intermediate Value Theorem

Ex.393The comparison test in practice

#comparison test #p-series

Convergence tests

Ex.394The derivative of an integral with variable bounds

#fundamental theorem of calculus #chain rule #limits

Riemann integral

Ex.395The largest rectangle in a semicircle

#optimisation #geometry #area

Optimisation and curve sketching

Ex.396The most economical can

#optimisation #cylinder #applications

Optimisation and curve sketching

Ex.397The ratio test

#ratio test #d'Alembert #factorials

Convergence tests

Ex.398Three ways to fail to be differentiable

#differentiability #corner #vertical tangent #oscillation

Differentiable functions

Ex.399Trigonometric integrals by linearisation

#trigonometric integrals #linearisation

Integration techniques

Ex.400Alternating sum of binomial coefficients

#alternating sum #binomial #zero

Binomial coefficients

Ex.401The binomial theorem

#binomial theorem #expansion #coefficients

Binomial coefficients

Ex.402A specific expansion

#expansion #binomial theorem #computation

Binomial coefficients

Ex.403Key binomial identities

#summary #identities #binomial

Binomial coefficients

Ex.404Sum of binomial coefficients

#sum identity #2^n #binomial theorem

Binomial coefficients

Ex.405Monotonicity of binomial row

#monotonicity #binomial #ratio

Binomial coefficients

Ex.406Even vs odd subsets

#even subsets #odd subsets #alternation

Binomial coefficients

Ex.407Pascal's identity

#Pascal identity #recurrence #binomial

Binomial coefficients

Ex.408Simplifying binomial quotients

#quotient #binomial #simplification

Binomial coefficients

Ex.409Recurrence for coefficients

#recurrence #Pascal #coefficients

Binomial coefficients

Ex.410Binomial as a quotient of factorials

#factorial #quotient #binomial

Binomial coefficients

Ex.411Sum over odd $k$

#odd k #sum #alternation

Binomial coefficients

Ex.412Sum over odd and even sizes

#sum parity #binomial #subsets

Binomial coefficients

Ex.413Sum over even $k$

#even k #sum #alternation

Binomial coefficients

Ex.414Sum of powers of 2

#power sum #2^n #binomial

Binomial coefficients

Ex.415Sum with $3^k$

#binomial #power sum #3^k

Binomial coefficients

Ex.416Countable sets

#countable #denumerable #enumeration

Countability and cardinality

Ex.417Hilbert's hotel

#Hilbert hotel #infinite #bijection

Countability and cardinality

Ex.418Infinite sets

#infinite sets #Dedekind #proper subset

Countability and cardinality

Ex.419$\mathbb{Z}$ is countable

#integers #countability #bijection

Countability and cardinality

Ex.420Summary of cardinality

#summary #cardinality

Countability and cardinality

Ex.421Midpoint and centroid in the complex plane

#midpoint #centroid #complex plane

Complex numbers

Ex.422Argument of a quotient

#argument #quotient #computation

Complex numbers

Ex.423Properties of the conjugate

#conjugate #properties #modulus

Complex numbers

Ex.424The cube roots of unity

#cube roots #unity #cyclic

Complex numbers

Ex.425De Moivre's formula

#De Moivre #powers #trigonometric form

Complex numbers

Ex.426Exponential form

#exponential form #Euler #polar

Complex numbers

Ex.427Geometric interpretation of operations

#complex plane #geometric interpretation #translation

Complex numbers

Ex.428Rotation via multiplication

#rotation #multiplication #geometric

Complex numbers

Ex.429Modulus and argument

#modulus #argument #trigonometric form

Complex numbers

Ex.430Modulus of a product

#modulus #product #quotient

Complex numbers

Ex.431A quadratic with complex roots

#quadratic #complex roots #conjugate roots

Complex numbers

Ex.432Product of complex numbers in trigonometric form

#trigonometric form #product #quotient

Complex numbers

Ex.433Solving $z^3=1$

#cube roots of unity #algebraic solution

Complex numbers

Ex.434A linear congruence

#linear congruence #inverse #solution

Ex.435Critical points of a polynomial

#critical points #polynomial #Hessian

Optimisation in several variables

Ex.436Summary of constrained optimization

#summary #constrained optimization

Optimisation in several variables

Ex.437Arrangements

#arrangements #ordered selection #permutations

Counting principles

Ex.438The binomial theorem

#binomial theorem #expansion #coefficients

Counting principles

Ex.439Pascal's identity

#Pascal identity #binomial #recurrence

Counting principles

Ex.440Expanding $(x-y)^n$

#binomial theorem #sign #expansion

Counting principles

Ex.441Choosing with repetition

#stars and bars #repetition #combinations

Counting principles

Ex.442Combinations

#combinations #unordered selection #binomial

Counting principles

Ex.443Choosing a mixed committee

#combinations #committee #product rule

Counting principles

Ex.444Counting injections

#injections #arrangements #permutations

Counting principles

Ex.445Permutations

#permutations #arrangements #n!

Counting principles

Ex.446Pigeonhole with balls

#pigeonhole #balls #boxes

Counting principles

Ex.447Subsets and their complements

#subsets #complement #bijection

Counting principles

Ex.448Round-robin tournament

#tournament #match counting #combination

Counting principles

Ex.449Determinant of a 3×3 matrix by Sarrus' rule

#determinant #Sarrus rule #3x3 matrices

Ex.450Determinant of an inverse

#inverse #determinant #formula

Ex.451Effect of row operations on the determinant

#row operations #determinant properties #multilinearity

Ex.452Determinant of a triangular matrix

#triangular matrix #determinant #diagonal

Ex.453Coprime integers

#coprime #gcd #Bézout

Divisibility and gcd

Ex.454GCD of three numbers

#gcd #three numbers #Euclidean

Divisibility and gcd

Ex.455Congruence classes modulo $n$

#congruences #residue classes #partition

Divisibility and gcd

Ex.456GCD and divisibility

#gcd #divisibility #equivalence

Divisibility and gcd

Ex.457LCM of three numbers

#lcm #three numbers #prime factorization

Divisibility and gcd

Ex.458The least common multiple

#lcm #gcd #product relation

Divisibility and gcd

Ex.459Every integer has a prime divisor

#primes #divisor #well-ordering

Divisibility and gcd

Ex.460Eigenvalues and eigenvectors of a 2×2 matrix

#eigenvalues #eigenvectors #characteristic polynomial

Eigenvalues and diagonalisation

Ex.461A 2×2 matrix with distinct eigenvalues

#eigenvalues #eigenvectors #distinct roots

Eigenvalues and diagonalisation

Ex.462An exponential equation

#exponential equation #substitution #quadratic

Equations and radicals

Ex.463A logarithmic equation

#logarithmic equation #domain #exponential

Equations and radicals

Ex.464A quadratic with complex roots

#quadratic #complex roots #discriminant

Equations and radicals

Ex.465Solving by factoring

#factoring #roots #quadratic

Equations and radicals

Ex.466A simple radical equation

#radical equation #squaring #extraneous roots

Equations and radicals

Ex.467A system by substitution

#system #substitution #nonlinear

Equations and radicals

Ex.468A trigonometric equation

#trigonometric equation #solving #periodicity

Equations and radicals

Ex.469The complement of a graph

#complement #graph #edges

Ex.470Connected components

#connectivity #connected components

Ex.471Paths and cycles

#path #cycle #walk

Ex.472Summary of graph theory

#summary #graph theory

#tree #connected acyclic #edges

Ex.474Leaves of a tree

#tree #leaves #degree

Ex.475Cyclic groups

#cyclic group #generator #examples

Ex.476Order of an element

#order #cyclic subgroup #element

Ex.477The subgroup criterion

#subgroup #criterion #nonempty

Ex.478Which subsets are subgroups?

#subgroup #examples #counterexamples

Ex.479The three-set inclusion-exclusion formula

#inclusion-exclusion #three sets #union

Inclusion-exclusion

Ex.480Counting complement via IE

#complement #IE #at least one

Inclusion-exclusion

Ex.481Derangements of $3$ elements

#derangements #$D_3$ #list

Inclusion-exclusion

Ex.482Derangements of $4$ elements

#derangements #computation #fixed point

Inclusion-exclusion

Ex.483Even vs odd subsets

#even subsets #odd subsets #IE

Inclusion-exclusion

Ex.484Exact intersection count

#IE #intersection #count

Inclusion-exclusion

Ex.485Summary of inclusion-exclusion

#summary #IE #formulas

Inclusion-exclusion

Ex.486Non-constant functions to $\{0,1\}$

#IE #functions #non-constant

Inclusion-exclusion

Ex.487Nested sets

#IE #nested #specific

Inclusion-exclusion

Ex.488Surjections to $\{0,1\}$

#surjections #IE #onto

Inclusion-exclusion

Ex.489The AM-GM inequality

#AM-GM #inequality #arithmetic and geometric means

Ex.490The triangle inequality

#triangle inequality #absolute value #real numbers

Ex.491Cauchy-Schwarz inequality

#Cauchy-Schwarz #inequality #inner product

Inner product spaces

Ex.492Orthogonal families are independent

#orthogonal family #linear independence #proof

Inner product spaces

Ex.493Between any two reals lies a rational

#density #rationals #betweenness

Ex.494√2 is algebraic

#algebraic #sqrt2 #minimal polynomial

Ex.495Proof by contradiction

#contradiction #irrationality #sqrt2

Proof techniques

Ex.496Proof by contraposition

#contrapositive #implication #odd

Proof techniques

Ex.497Contrapositive with primes

#contrapositive #prime #divisibility

Proof techniques

Ex.498Converse, inverse, contrapositive

#implication #converse #contrapositive

Proof techniques

Ex.499Bernoulli's inequality by induction

#induction #Bernoulli #inequality

Ex.500Sum of squares by induction

#induction #sum of squares #integers

Ex.501Proof by cases

#proof by cases #parity #absolute value

Proof techniques

Ex.502A parity case analysis

#cases #parity #square

Proof techniques

Ex.503A tautology

#propositional logic #tautology #truth table

Proof techniques

Ex.504Negating quantifiers

#quantifiers #negation #logic

Proof techniques

Ex.505Summary of logic and proof techniques

#summary #proof #logic

Proof techniques

Ex.506Determinant of a 3×3 matrix by Sarrus' rule

#determinant #Sarrus rule #3x3 matrices

Matrices and linear systems

Ex.507Inverse of a 2×2 matrix

#inverse #determinant #2x2 matrices

Matrices and linear systems

Ex.508Matrix multiplication

#matrix multiplication #non-commutativity

Matrices and linear systems

Ex.509Powers of a diagonal matrix

#diagonal matrix #powers #induction

Matrices and linear systems

Ex.510Properties of the trace

#trace #cyclic property #similarity invariance

Matrices and linear systems

Ex.511Properties of the transpose

#transpose #properties #matrix operations

Matrices and linear systems

Ex.512Cauchy sequences

#Cauchy #sequences #metric

Metric and normed spaces

Ex.513Continuity in metric spaces

#continuity #epsilon-delta #metric

Metric and normed spaces

Ex.514Convergence in metric spaces

#convergence #sequences #metric

Metric and normed spaces

Ex.515Classical metrics on $\mathbb{R}^n$

#Euclidean #Manhattan #sup norm

Metric and normed spaces

Ex.516Normed spaces

#norm #definition #axioms

Metric and normed spaces

Ex.517Separable integrand on a square

#separability #product of integrals #square

Multiple integrals

Ex.518Summary of multiple integrals

#summary #multiple integrals

Multiple integrals

Ex.519Double integral over a triangular domain

#triangle #double integral #bounds

Multiple integrals

Ex.520Double integral under a parabola

#double integral #type-I domain #parabola

Multiple integrals

Ex.521A symmetric triangle integral

#symmetry #triangle #double integral

Multiple integrals

Ex.522Directional derivative

#directional derivative #gradient #direction

Partial derivatives

Ex.523Euler's theorem for homogeneous functions

#homogeneous functions #Euler #partial derivatives

Partial derivatives

Ex.524The gradient

#gradient #vector #direction

Partial derivatives

Ex.525Among $n+1$ from $2n$, two consecutive

#pigeonhole #consecutive #betweenness

Pigeonhole principle

Ex.526Consecutive integers

#pigeonhole #consecutive integers #pairs

Pigeonhole principle

Ex.527Difference divisible by $n$

#pigeonhole #difference #divisibility

Pigeonhole principle

Ex.528Pigeonhole with numbers

#pigeonhole #numbers #residue

Pigeonhole principle

Ex.529More socks

#pigeonhole #socks #colors

Pigeonhole principle

Ex.530Summary of pigeonhole techniques

#summary #pigeonhole #techniques

Pigeonhole principle

Ex.531Generalized pigeonhole with $100$ people

#generalized pigeonhole #ceil #months

Pigeonhole principle

Ex.532Balls in boxes

#generalized pigeonhole #ceil #balls

Pigeonhole principle

Ex.533Proof of the generalized pigeonhole principle

#generalized pigeonhole #proof #ceil

Pigeonhole principle

Ex.534A pair with difference of $n$

#pigeonhole #difference #exact value

Pigeonhole principle

Ex.535Pairs summing to $2n+1$

#pigeonhole #sum #pairs

Pigeonhole principle

Ex.536Divisibility by $X-a$

#factor theorem #divisibility #roots

Ex.537Euclidean division

#Euclidean division #quotient #remainder

Ex.538The factor theorem

#factor theorem #roots #divisibility

Ex.539Horner's scheme

#Horner scheme #evaluation #algorithm

Ex.540Vieta's formulas for a quadratic

#Vieta #sum and product of roots #quadratic

Ex.541Consecutive primes and coprimality

#consecutive primes #coprime #primorial

Ex.542Coprimality with primes

#coprime #prime #divisibility

Ex.543The fundamental theorem of arithmetic

#fundamental theorem #factorization #uniqueness

Ex.544GCD and LCM via prime factorization

#gcd #lcm #prime factorization

Ex.545Primes in GCD and LCM

#gcd #lcm #prime factorization

Ex.546Goldbach's conjecture (statement and verification)

#Goldbach conjecture #even numbers #primes

Ex.547Euclid's proof of infinitely many primes

#Euclid #contradiction #primes

Ex.548Iterated GCD

#gcd #iteration #associativity

Ex.549Prime factorization and divisors

#prime factorization #divisors #exponents

Ex.550Characterization of primes

#primes #characterization #divisors

Ex.551Using the sieve to test primality

#sieve #primality #application

Ex.552The sieve of Eratosthenes

#sieve #Eratosthenes #primes

Ex.553Smallest prime divisor

#smallest divisor #compositeness #sqrt bound

Ex.554The smallest prime factor

#smallest prime factor #compositeness #bounds

Ex.555Trial division primality test

#trial division #primality test #algorithm

Ex.556Twin primes

#twin primes #examples #conjecture

Ex.557Polar form of a quadratic form

#polar form #symmetric bilinear form #quadratic form

Quadratic forms

Ex.558Bounded quantifiers

#bounded quantifiers #notation #domain

Quantifiers and negation

Ex.559Quantifier over a finite set

#finite quantifier #exists #disjunction

Quantifiers and negation

Ex.560Negating an existential statement

#negation #existential #exists

Quantifiers and negation

Ex.561Negating a universal statement

#negation #universal #forall

Quantifiers and negation

Ex.562Summary of quantifiers and negation

#summary #quantifiers #negation

Quantifiers and negation

Ex.563The uniqueness quantifier

#uniqueness #exists unique #quantifier

Quantifiers and negation

Ex.564Refuting a universal statement

#refutation #counterexample #forall

Quantifiers and negation

Ex.565Antisymmetric relations

#antisymmetric #relation #definition

Ex.566Equivalence classes

#equivalence class #congruence #quotient

Ex.567Equivalence relations

#equivalence relation #reflexive #symmetric #transitive

Ex.568Partial orders

#partial order #poset #subsets

#field #examples #invertible elements

Rings and fields

Ex.570Integral domains

#integral domain #examples #cancellation

Rings and fields

Ex.571Zero divisors

#zero divisors #examples #rings

Rings and fields

Ex.572Cardinality of finite sets

#cardinality #finite sets #inclusion-exclusion

Ex.573Cartesian product

#Cartesian product #ordered pair #cardinality

Ex.574Complement laws

#complement #De Morgan #laws

Ex.575Distributive laws for sets

#distributivity #union #intersection

Ex.576The power set

#power set #subsets #cardinality

Ex.577Relations and their graphs

#relation #binary relation #graph

Ex.578Symmetric difference

#symmetric difference #XOR #sets

Ex.579The binomial expansion of √(1 + x)

#Taylor expansion #binomial series #square root

Taylor expansions

Ex.580The Taylor expansion of arctan by integrating

#Taylor expansion #integration of expansions #arctangent

Taylor expansions

Ex.581Closed sets

#closed sets #complement #definition

Topology of ℝ

Ex.582Closed intervals are closed

#closed sets #closed intervals #complement

Topology of ℝ

Ex.583Finite intersections of open sets

#intersections #open sets #finite

Topology of ℝ

Ex.584Arbitrary unions of open sets

#unions #open sets #arbitrary

Topology of ℝ

Ex.585Boundary of an open set

#boundary #open set #disjointness

Topology of ℝⁿ

Ex.586Boundary of a ball

#boundary #sphere #ball

Topology of ℝⁿ

Ex.587Interior, closure, boundary

#interior #closure #boundary

Topology of ℝⁿ

Ex.588Limit points in $\mathbb{R}^n$

#limit points #sequences #closure

Topology of ℝⁿ

Ex.589Stability of open and closed sets

#unions #intersections #stability

Topology of ℝⁿ

Ex.590Recognizing bases

#basis #dimension #canonical basis

Vector spaces and subspaces

Ex.591The divergence

#divergence #vector field #computation

Vector calculus

Ex.592Closed paths in conservative fields

#closed path #conservative #zero integral

Vector calculus

Ex.593Line integral of a scalar field

#line integral #scalar field #parametrization

Vector calculus

Ex.594Linear independence

#linear independence #definition #examples

Vector spaces and subspaces

Ex.595Span of a set of vectors

#span #linear combinations #generating set

Vector spaces and subspaces

Ex.596The subspace criterion

#subspace criterion #examples

Vector spaces and subspaces

Ex.597Which subsets are subspaces?

#subspace criterion #counterexamples

Vector spaces and subspaces

Ex.598√2 + √3 is irrational

#rational root theorem #irrationality #minimal polynomial

Ex.599Power sums from the first three

#Vieta #Newton sums #symmetric polynomials

Ex.600The remainder of a huge power

#polynomial division #remainder #roots of unity

Ex.601A block of consecutive terms divisible by n

#pigeonhole #prefix sums #remainders

Pigeonhole principle

Ex.602Dominoes on a mutilated chessboard

#colouring #tilings #dominoes

Counting principles

Ex.603Replacing two numbers by their difference

#invariant #parity #board games

Counting principles

Ex.604Regions cut by n lines

#induction #counting #lines in the plane

Counting principles

Ex.605A combinatorial proof that Σ C(n,k)² = C(2n,n)

#double counting #binomial identities #Vandermonde

Binomial coefficients

Ex.606Two people with the same number of friends

#pigeonhole #graphs #degrees

Pigeonhole principle

Ex.607Two take-away games

#games #winning positions #pairing strategy #Nim

Counting principles

Ex.608The expected number of fixed points of a random permutation

#linearity of expectation #indicator variables #permutations

Counting principles

Ex.609The point minimising the sum of distances to the vertices of a convex quadrilateral

#triangle inequality #extremal #convex quadrilateral

Ex.610Minimising 1/a + 4/b + 9/c on a + b + c = 1

#Cauchy-Schwarz #constrained minimum #equality case

Ex.611Nesbitt's inequality

#Nesbitt #Cauchy-Schwarz #Titu's lemma

Ex.612Two inequalities from AM-GM on pairs

#AM-GM #symmetric inequalities #warm-up

Ex.613a²/b + b²/c + c²/a ≥ a + b + c

#Titu's lemma #Cauchy-Schwarz #cyclic sums

Ex.614a³ + b³ + c³ ≥ a²b + b²c + c²a

#rearrangement inequality #AM-GM #cyclic sums

Ex.615Weighted AM-GM: how to choose the splitting

#weighted AM-GM #optimisation #equality case

Ex.616A function with zero derivative is constant

#consequence #constant function #zero derivative

Mean Value Theorem

Ex.617Sign of the derivative and monotonicity

#monotonicity #derivative sign #consequence

Mean Value Theorem

Ex.618n + S(n) + S(S(n)) = 60

#digit sum #congruence modulo 9 #bounding

Ex.6198k + 7 is not a sum of three squares

#squares modulo 8 #representation by squares

Ex.620Counting divisors and the hundred lockers

#divisor function #perfect squares #pairing

Divisibility and gcd

Ex.621x² + y² = 3z² has no non-trivial solution

#infinite descent #squares modulo 3 #Diophantine equations

Diophantine equations

Ex.622x² − y² = 2024

#Diophantine equations #factoring #parity #divisors

Diophantine equations

Ex.623gcd(n! + 1, (n + 1)! + 1) = 1

#gcd #divisibility #factorials

Divisibility and gcd

Ex.62413 divides 2⁷⁰ + 3⁷⁰

#Fermat's little theorem #modular exponentiation

Divisibility and gcd

Ex.625When can 2ⁿ + 1 or 2ⁿ − 1 be prime?

#factorisation #primes #Fermat primes #Mersenne primes

Ex.626Inclusion and complement

#inclusion #complement #direct proof #contradiction

Ex.627Inclusion and union with the complement

#inclusion #complement #equivalence

Ex.628Cardinality of the power set

#power set #cardinality #induction

Ex.629Continuity of the determinant

#determinant #continuity #polynomial

Topology of Mn(ℝ)

Ex.630Convergence in $M_n(\mathbb{R})$ via entries

#matrix convergence #entrywise convergence #sequence of matrices

Topology of Mn(ℝ)

Ex.631Summary of the topology of $M_n(\mathbb{R})$

#summary #matrix topology #M_n(R)

Topology of Mn(ℝ)

Ex.632Cassini's identity and sums of Fibonacci squares

#Fibonacci numbers #matrices #determinant #induction

Equations and radicals

Ex.633How many real roots does x⁵ − 5x + a have?

#polynomials #derivative #number of roots #parameter

Ex.634Three telescoping sums

#telescoping #factorials #partial fractions

Equations and radicals

Ex.635A subtraction game with moves 1, 3 and 4

#combinatorial games #losing positions #induction #periodicity

Counting principles

Ex.636How many colours does a graph with m edges need?

#graph colouring #chromatic number #extremal

Ex.637Six people: three friends or three strangers

#Ramsey theory #pigeonhole #colouring

Pigeonhole principle

Ex.638Tiling a 2 × n board with dominoes and squares

#recurrence #tilings #generating functions

Counting principles

Ex.639Cauchy's equation on the rationals

#Cauchy equation #additive function #rationals

Functional equations

Ex.640Stewart's theorem and its consequences

#Stewart's theorem #medians #angle bisector #law of cosines

Ex.641Two classical locus problems

#locus #coordinates #perpendicular #circle with a given diameter

Analytic geometry

Ex.642A chain of squares: a⁴ + b⁴ + c⁴ ≥ abc(a + b + c)

#sum of squares #SOS #real numbers

Ex.643When is 1! + 2! + … + n! a perfect square?

#squares modulo n #factorials #Diophantine

Diophantine equations

Ex.644Writing 1 as a sum of unit fractions

#Egyptian fractions #bounding #Diophantine

Diophantine equations

Ex.645The sum of φ(d) over the divisors of n

#Euler totient #divisors #counting #fractions

Divisibility and gcd

Ex.646A 2×2 matrix with positive entries has a positive eigenvector

#intermediate value theorem #Perron-Frobenius #eigenvector

Applications and the case n = 2

Ex.647Convexity of Î»_max and the spectral norm on symmetric matrices

#convexity #Rayleigh quotient #norm #symmetric matrices

Applications and the case n = 2

Ex.648A discrete cyclic subgroup of the general linear group

#discrete subgroup #closed set #powers of a matrix

Density and closures

Ex.649Closure of the PSD matrices of rank k

#positive semidefinite #rank #closure

Density and closures

Ex.650Invertible rational matrices are dense in the general linear group

#rational matrices #density #open set

Density and closures

Ex.651Real-diagonalisable matrices are not dense

#diagonalisation #non-real eigenvalues #continuity of roots

Density and closures

Ex.652Three open dense subsets of the space of matrices

#density #discriminant #principal minors #polynomials

Density and closures

Ex.653The closure of a subgroup of GL_n is a subgroup

#topological group #closure #subgroup

Exponential, logarithm and group topology

Ex.654The derivative of the matrix exponential and the one-parameter group property

#matrix exponential #derivative #homomorphism

Exponential, logarithm and group topology

Ex.655The exponential is not injective: rotations as exponentials

#matrix exponential #rotation #kernel

Exponential, logarithm and group topology

Ex.656Existence of a closest matrix of rank at most r

#low rank approximation #compactness #coercivity

Rank, spectrum and norms

Ex.657Singular matrices have measure zero

#Lebesgue measure #zero set of a polynomial #Fubini #determinant

Rank, spectrum and norms

Ex.658The singular value map is continuous and onto the ordered cone

#singular values #Weyl inequality #continuity #surjectivity

Rank, spectrum and norms

Ex.659Strictly diagonally dominant matrices are invertible

#diagonal dominance #Gershgorin #open sets #invertibility

Rank, spectrum and norms

Ex.660The numerical range of a real matrix is an interval

#Rayleigh quotient #connectedness #compactness #symmetric part

Rank, spectrum and norms

Ex.661The Moore-Penrose pseudo-inverse is discontinuous

#pseudo-inverse #discontinuity #rank #singular value decomposition

Rank, spectrum and norms

Ex.662Equivalence between $\sum u_n$ and $\sum\ln(1+u_n)$

#equivalence #logarithm #ratio test #convergence

Convergence tests

Ex.663The Frobenius norm is submultiplicative

#Frobenius norm #Cauchy-Schwarz #submultiplicativity

Norms and metric structure

Ex.664Operator norm: submultiplicativity and comparison with Frobenius

#operator norm #submultiplicativity #Frobenius norm

Norms and metric structure

Ex.665Operator norms of three basic matrices

#operator norm #nilpotent #diagonal #rotation

Norms and metric structure

Ex.666The max-entry norm is not submultiplicative

#max norm #submultiplicativity #counterexample

Norms and metric structure

Ex.667Orthogonal invariance of the Frobenius and operator norms

#orthogonal matrices #isometry #trace cyclicity

Norms and metric structure

Ex.668Operator norm of a symmetric matrix

#symmetric matrices #spectral theorem #operator norm

Norms and metric structure

Ex.669The diameter of the orthogonal group

#orthogonal group #diameter #triangle inequality

Norms and metric structure

Ex.670Open ball, closed ball and sphere in the operator norm

#operator norm #ball #closure #boundary

Open, closed, interior, boundary

Ex.671Positive definite matrices: open cone with closure PSD

#positive definite #cones #eigenvalues #interior

Open, closed, interior, boundary

Ex.672PSD matrices have empty interior in all of M_n

#positive semidefinite #symmetric matrices #interior

Open, closed, interior, boundary

Ex.673Nilpotent matrices and idempotents are closed

#nilpotent #idempotent #algebraic sets #nowhere dense

Open, closed, interior, boundary

Ex.674GL_n is dense and its boundary is the singular set

#invertible matrices #density #boundary #determinant

Open, closed, interior, boundary

Ex.675Matrices with no zero entry: open and dense

#nowhere dense #hyperplanes #open dense sets

Open, closed, interior, boundary

Ex.676Entrywise nonnegative matrices: a closed cone with interior

#cones #convexity #interior #nonnegative matrices

Open, closed, interior, boundary

Ex.677Conjugation is continuous

#conjugation #composition of continuous maps #similarity

Continuity of matrix maps

Ex.678Lipschitz estimates for the norm and for inversion

#Lipschitz #inversion #resolvent identity #operator norm

Continuity of matrix maps

Ex.679Inversion is not uniformly continuous

#uniform continuity #inversion #singular matrices #counterexample

Continuity of matrix maps

Ex.680Matrices of negative determinant are not squares

#squares #determinant sign #surjectivity

Continuity of matrix maps

Ex.681The nearest symmetric matrix is the symmetric part

#symmetric matrices #skew-symmetric #orthogonal projection #Frobenius norm

Continuity of matrix maps

Ex.682The characteristic polynomial map and its section

#characteristic polynomial #companion matrix #continuous section

Continuity of matrix maps

Ex.683The nilpotent cone is closed but not compact

#nilpotent matrices #fibre #Cayley-Hamilton #unbounded set

Continuity of matrix maps

Ex.684A path from positive to negative determinant hits a singular matrix

#intermediate value theorem #determinant #paths #connectedness

Continuity of matrix maps

Ex.685Complement of a subspace

#hyperplane #codimension #connectedness #linear subspace

Ex.686The complement of O(n) is path connected

#orthogonal group #complement #singular values #path connected

Ex.687Hurwitz (stable) matrices form a connected set

#Hurwitz matrices #stability #eigenvalues #linear deformation

Ex.688Level sets of the determinant

#determinant #level sets #SL_n #scaling

Ex.689The general linear group is path connected

#polar decomposition #general linear group #path connected

Ex.690Matrices with real positive spectrum

#eigenvalues #linear deformation #path connected

Ex.691O(n) has exactly two components

#orthogonal group #components #reflections #homeomorphism

Ex.692Spectral radius balls and star-shaped sets

#spectral radius #star-shaped #norm ball

Ex.693Nilpotent matrices and idempotents of fixed rank

#nilpotent matrices #idempotents #conjugacy orbits #projections

Ex.694The idempotents 0 and I are isolated points

#idempotents #projections #isolated points #trace and rank

Applications and the case n = 2

Ex.695Square-zero matrices form a path connected set

#nilpotent matrices #star-shaped sets #path connectedness #cones

Applications and the case n = 2

Ex.696The operator-norm distance between two rotations is a chord length

#M2(R) #rotations #operator norm #chord length

Applications and the case n = 2

Ex.697Powers of a matrix

#diagonalisation #matrices

Eigenvalues and diagonalisation

Ex.698A recursive sequence converging to 2

#sequences #monotone convergence

Ex.699AM-GM for three numbers

#inequalities #AM-GM

Ex.700Convergence of the series of 1/n²

#series #comparison

Convergence tests

Ex.701Derangements of four objects

#inclusion-exclusion #permutations

Inclusion-exclusion

Ex.702Divergence of the harmonic series

#series #divergence

Convergence tests

Ex.703Inradius of a right triangle

#incircle #right triangle

Ex.704Powers of a 2×2 matrix with eigenvalues 1 and 2

#diagonalisation #matrices

Eigenvalues and diagonalisation

Ex.705The centroid of a triangle

#vectors #medians

Ex.70630 divides n⁵ − n

#divisibility #factorisation

Divisibility and gcd

Ex.707Cosine to the power 1/x²

#exponential form #Taylor expansion #indeterminate forms

Ex.708Difference of two infinite reciprocals

#infinity minus infinity #Taylor expansion

Ex.709Epsilon-delta for x squared

#epsilon-delta #definition of limit #restricting delta

Ex.710The gap between two exponentials at infinity

#exponential #limits at infinity #change of variable

Ex.711The logarithm is slower than any power

#growth comparison #logarithm #inequalities

Ex.712(1 + sin x)^(1/x)

#exponential form #indeterminate forms #composition

Ex.713Polynomials versus the exponential

#growth comparison #exponential #squeeze theorem

Ex.714sin x − x cos x over x cubed

#Taylor expansion #little-o notation #cancellation

Ex.715sin(1/x) has no limit at 0

#sequences #non-existence of limits

Ex.716Tangent minus sine over x cubed

#trigonometric limits #factoring

Ex.717x ln x at zero, and x^x

#indeterminate forms #change of variable #logarithm

Ex.718x sin x has no limit at infinity

#sequences #non-existence of limits #infinite limits

Ex.719x times the Dirichlet function

#squeeze theorem #pathological functions #continuity

Ex.720A continuous function with rational values is constant

#intermediate value theorem #rational numbers #proof by contradiction

Intermediate Value Theorem

Ex.721Fixed point and a derivative bounded by one

#fixed points #mean value theorem #intermediate value theorem

Mean Value Theorem

Ex.722A recurrent sequence and its fixed point

#fixed point #contraction #mean value theorem

Ex.723A reduction formula

#reduction formula #integration by parts #recurrence

Integration techniques

Ex.724Cancellation revealed by a Taylor expansion

#Taylor expansion #equivalents #cancellation

Convergence tests

Ex.725A telescoping sum that proves ζ(2) converges

#telescoping #comparison #zeta function

Convergence tests

Ex.726Absolute, conditional or no convergence

#absolute convergence #alternating series #p-series

Convergence tests

Ex.727Solving an integral equation

#integral equations #fundamental theorem #differential equation

Riemann integral

Ex.728Completing the square before integrating

#rational functions #completing the square #arctangent

Integration techniques

Ex.729An integral singular at both endpoints

#improper integrals #substitution #arcsine

Improper integrals

Ex.730Binet's formula for the Fibonacci numbers

#Fibonacci #golden ratio #linear recurrence #limits

Ex.731Classifying a critical point when f″(0) = 0

#local extrema #Taylor expansion #second derivative test

Taylor expansions

Ex.732Comparing the harmonic sum with the logarithm

#integral comparison #harmonic series #monotonicity

Riemann integral

Ex.733Lagrange multipliers on a circle

#Lagrange #multipliers #equality constraint

Optimisation in several variables

Ex.734Optimisation under a linear constraint

#Lagrange #linear constraint #minimum

Optimisation in several variables

Ex.735Convergence to the golden ratio

#golden ratio #contraction #fixed point

Ex.736cos x = 1/x has infinitely many solutions

#intermediate value theorem #infinitely many solutions

Intermediate Value Theorem

Ex.737Counting solutions depending on a parameter

#intermediate value theorem #monotonicity #parameters

Intermediate Value Theorem

Ex.738Integrating by parts twice and coming back

#integration by parts #exponential #trigonometric

Integration techniques

Ex.739Darboux sums of the Dirichlet function

#Riemann integrability #Darboux sums #density

Riemann integral

Ex.740A divisibility identity for a sum

#divisibility #linearity #sum

Divisibility and gcd

Ex.741Bézout identity for 1001 and 315

#Euclidean algorithm #Bézout identity

Divisibility and gcd

Ex.742When are $2n$ and $3n+1$ coprime?

#divisibility #linear combination #coprimality

Divisibility and gcd

Ex.743Coprime powers remain coprime

#coprime #powers #Bézout

Divisibility and gcd

Ex.744If $a,b$ are coprime, so are $a+b$ and $ab$

#coprime #sum #product

Divisibility and gcd

Ex.745Divisibility of $a^n-b^n$

#divisibility #factorization #powers

Divisibility and gcd

Ex.746Divisors of a product

#divisibility #coprime #product divisibility

Divisibility and gcd

Ex.747Why the Euclidean algorithm works

#Euclidean algorithm #proof #division

Divisibility and gcd

Ex.748The Euclidean algorithm and Bézout identity

#Euclidean algorithm #Bézout identity #divisibility computation

Divisibility and gcd

Ex.749When is $(6n+12)/(2n+1)$ an integer?

#divisibility #fraction #case analysis

Divisibility and gcd

Ex.750An incompatible congruence system

#Chinese remainder #compatibility condition

Ex.751When does $n-4$ divide $3n-17$?

#divisibility #linear combination #case analysis

Divisibility and gcd

Ex.752When is the number of divisors odd?

#divisors #perfect squares #counting

Divisibility and gcd

Ex.753A divisibility recurrence

#divisibility #recurrence #Euclidean algorithm

Divisibility and gcd

Ex.754The smallest prime divisor

#primes #smallest divisor #compositeness

Divisibility and gcd

Ex.755Which is larger, e^π or π^e?

#optimisation #inequalities #logarithm

Optimisation and curve sketching

Ex.756tan x = x on each branch

#intermediate value theorem #tangent #monotonicity

Intermediate Value Theorem

Ex.757A linear first-order equation with polynomial right-hand side

#linear equation #homogeneous solution #particular solution

First-order equations

Ex.758Newton's law of cooling

#Newton cooling #separable equation #modelling

First-order equations

Ex.759Variation of constants

#variation of constants #linear equation #integration

First-order equations

Ex.760Functions whose derivative is proportional to themselves

#exponential function #differential equation #functional equations

Differentiable functions

Ex.761Heron's method for √2

#Newton's method #quadratic convergence #square roots

Ex.762If f′ tends to ℓ, then f(x+1) − f(x) tends to ℓ

#mean value theorem #limits at infinity

Mean Value Theorem

Ex.763Integrating ln and arctan by parts

#integration by parts #logarithm #arctangent

Integration techniques

Ex.764L'Hôpital's rule, twice

#L'Hôpital's rule #indeterminate forms

Differentiable functions

Ex.765A diagonalizable system

#diagonalization #eigenvalues #eigenvectors

Ex.766A system with three distinct eigenvalues

#eigenvalues #eigenvectors #general solution

Ex.767Derivative of the matrix exponential

#matrix exponential #derivative #series

Ex.768Properties of the matrix exponential

#matrix exponential #commuting matrices #derivative

Ex.769Double integral in polar coordinates

#polar coordinates #double integral #Jacobian

Multiple integrals

Ex.770Triple integral over a simplex

#triple integral #iterated integration #simplex

Multiple integrals

Ex.771The chain rule for a composition

#chain rule #composition #directional derivative

Partial derivatives

Ex.772Power series of ln(1+x) and arctan x

#integration term by term #Leibniz series #ln 2

Ex.773Radius of convergence, four examples

#radius of convergence #ratio test #endpoints

Ex.774The sine of an alternating term

#alternating series #odd function #conditional convergence

Convergence tests

Ex.775Solving y″ = −y with a power series

#differential equations #power series #cosine #recurrence

Ex.776Summing power series in closed form

#power series #differentiation term by term #closed form

Ex.777Supremum, infimum, limsup and liminf

#supremum #limsup #liminf #bounded sequences

Ex.778Approximating e with a guaranteed error

#Taylor-Lagrange #error bound #number e

Taylor expansions

Ex.779Alternating series and the error bound

#alternating series test #remainder #error estimate

Convergence tests

Ex.780The area of an ellipse

#trigonometric substitution #area #ellipse

Integration techniques

Ex.781The Cauchy–Schwarz inequality for integrals

#inequalities #Cauchy-Schwarz #quadratic polynomial

Riemann integral

Ex.782The closest point on a parabola

#optimisation #distance #parabola

Optimisation and curve sketching

Ex.783Differentiating under the integral sign

#parameter integrals #Gamma function #factorial

Improper integrals

Ex.784The hierarchy of growth

#growth comparison #ratio test #factorial

Ex.785The integral test and Bertrand series

#integral test #Bertrand series #logarithm

Convergence tests

Ex.786Using the substitution x → a + b − x

#symmetry #substitution #trigonometric integrals

Integration techniques

Ex.787The limit comparison test

#limit comparison #equivalents #Taylor expansion

Convergence tests

Ex.788The limit of the derivative theorem

#mean value theorem #derivative at a point #extension

Mean Value Theorem

Ex.789The n-th derivative of 1/(x² − 1)

#higher derivatives #partial fractions

Differentiable functions

Ex.790The n-th derivative of x² e^x

#higher derivatives #Leibniz formula #induction

Differentiable functions

Ex.791The root test

#root test #Cauchy #exponential limits

Convergence tests

Ex.792Three cases of order-two recurrences

#linear recurrence #double root #complex roots

Ex.793Two alternating series with logarithms

#alternating series test #monotonicity #conditional convergence

Convergence tests

Ex.794Line integral of a differential form

#line integral #differential form #parametrization

Vector calculus

Ex.795When L'Hôpital's rule gives no answer

#L'Hôpital's rule #counterexamples #squeeze theorem

Differentiable functions

Ex.796An equivalent of e^(sin x) − e^(tan x)

#equivalents #expansions #difference of functions

Taylor expansions

Ex.797A one-sided bound for √(1 + x) from the Lagrange remainder

#Taylor-Lagrange #inequality #remainder sign

Taylor expansions

Ex.798Computing ∫₀¹ sin(x²) dx with a series

#integration of expansions #numerical approximation #alternating series

Taylor expansions

Ex.799Parity of binomial coefficients

#parity #binomial #Lucas

Binomial coefficients

Ex.800A combinatorial proof

#combinatorial proof #identity #binomial

Binomial coefficients

Ex.801Combinatorial proof

#combinatorial proof #binomial #choice

Binomial coefficients

Ex.802Compositions via binomial coefficients

#compositions #binomial #counting

Binomial coefficients

Ex.803Expansion of a large binomial

#binomial #expansion #coefficients

Binomial coefficients

Ex.804Telescoping with binomials

#telescoping #binomial #series

Binomial coefficients

Ex.805The hockey-stick identity

#hockey stick #sum identity #Pascal

Binomial coefficients

Ex.806An inequality with binomials

#inequality #binomial #bound

Binomial coefficients

Ex.807Counting with binomials

#lattice #paths #binomial

Binomial coefficients

Ex.808The largest binomial coefficient

#maximum binomial #middle coefficient #parity

Binomial coefficients

Ex.809Multinomial identity

#multinomial #identity #coefficient

Binomial coefficients

Ex.810Generalized binomial coefficients

#generalized binomial #negative exponent #series

Binomial coefficients

Ex.811Polynomial expansion with binomials

#polynomial #coefficient #binomial

Binomial coefficients

Ex.812Coefficients of a power

#coefficient #expansion #binomial

Binomial coefficients

Ex.813Application to sums

#binomial #application #sum

Binomial coefficients

Ex.814Sum with two powers

#sum #binomial #powers

Binomial coefficients

Ex.815Vandermonde's identity

#Vandermonde #convolution #binomial

Binomial coefficients

Ex.816Weighted sum of binomial coefficients

#weighted sum #binomial #derivative

Binomial coefficients

Ex.817The cardinal $\aleph_0$

#aleph zero #notation #cardinality

Countability and cardinality

Ex.818The cardinality of the continuum

#continuum #cardinality #2^aleph0

Countability and cardinality

Ex.819$\mathbb{Q}$ is dense but countable

#rationals #density #countability

Countability and cardinality

Ex.820Cantor's diagonal argument

#diagonal argument #Cantor #uncountability

Countability and cardinality

Ex.821Comparing cardinalities

#comparison #injection #bijection

Countability and cardinality

Ex.822$\mathbb{N}^2$ is countable

#N squared #diagonal #bijection

Countability and cardinality

Ex.823Countable product

#product #countable #product rule

Countability and cardinality

Ex.824$\mathbb{Q}$ is countable

#rationals #countability #surjection

Countability and cardinality

Ex.825Argument of a product

#argument #product #additive property

Complex numbers

Ex.826Conjugate roots of real polynomials

#conjugate roots #real polynomials #complex

Complex numbers

Ex.827Cube roots of a complex number

#cube roots #complex #De Moivre

Complex numbers

Ex.828Linearizing powers of cosine

#De Moivre #linearization #cos

Complex numbers

Ex.829De Moivre's formula by induction

#De Moivre #induction #proof

Complex numbers

Ex.830A circle given by a modulus equation

#modulus #circle #locus

Complex numbers

Ex.831The modulus equation $|z-a|=|z-b|$

#perpendicular bisector #modulus #locus

Complex numbers

Ex.832Triangle inequality for complex numbers

#triangle inequality #modulus #complex

Complex numbers

Ex.833A locus defined by a modulus equation

#locus #modulus #geometric interpretation

Complex numbers

Ex.834The $n$-th roots of unity

#roots of unity #De Moivre #cyclic group

Complex numbers

Ex.835The perpendicular bisector

#locus #perpendicular bisector #modulus

Complex numbers

Ex.836Square roots of a complex number

#square roots #complex #algebraic method

Complex numbers

Ex.837Computing roots with De Moivre

#nth roots #De Moivre #formula

Complex numbers

Ex.838Sum of the $n$-th roots of unity

#roots of unity #sum #geometric series

Complex numbers

Ex.839Linearizing $\sin^4\theta$

#linearization #sin #De Moivre

Complex numbers

Ex.840A fast series for ln 2

#logarithm #series #numerical approximation #speed of convergence

Taylor expansions

Ex.841Coprimality via congruences

#coprime #congruence #linear combination

Ex.842A CRT application

#CRT #application #calendar

Ex.843The Chinese remainder theorem (two moduli)

#Chinese remainder theorem #CRT #coprime moduli

Ex.844Euler's theorem

#Euler theorem #totient #modular exponentiation

Ex.845Computing $\varphi(n)$

#Euler totient #computation #prime factorization

Ex.846Generalization of Fermat

#Fermat #generalization #coprime

Ex.847Fermat's little theorem

#Fermat #prime #modular exponentiation

Ex.848Halving in modular arithmetic

#halving #modular arithmetic #odd modulus

Ex.849Finding modular inverses

#modular inverse #extended Euclidean #algorithm

Ex.850Inverse via Fermat

#Fermat #inverse #modular arithmetic

Ex.851Solvability of $ax\equiv b\pmod n$

#linear congruence #gcd criterion #number of solutions

Ex.852Applications of the modular inverse

#modular inverse #application #linear congruence

Ex.853Power cycles modulo $n$

#power cycle #periodicity #modulus

Ex.854A system of congruences

#system #CRT #linear congruences

Ex.855The order of an element mod $n$

#order #multiplicative group #divisor

Ex.856Computing large powers mod $n$

#modular exponentiation #square-and-multiply #fast power

Ex.857Boundary classification

#boundary #classification #circle

Optimisation in several variables

Ex.858Distance from point to plane

#Lagrange #distance #plane

Optimisation in several variables

Ex.859Two critical points and their nature

#critical points #saddle #maximum

Optimisation in several variables

Ex.860Lagrange multipliers on a circle

#Lagrange #circle #extrema

Optimisation in several variables

Ex.861Optimization under a linear constraint

#Lagrange #linear constraint #minimum

Optimisation in several variables

Ex.862Maximizing a product

#Lagrange #product #constraint

Optimisation in several variables

Ex.863A unique critical point that is a saddle

#saddle #unique critical #Hessian

Optimisation in several variables

Ex.864A discontinuous function with bounded values

#discontinuity #bounded #counterexample

Ex.865Discontinuity via two paths

#discontinuity #paths #limits

Ex.866Norm and distance are continuous

#norm #distance #continuity

Ex.867Continuity of a piecewise function

#piecewise #continuity #xy

Ex.868Continuity via polar coordinates

#polar #continuity #bound

Ex.869Continuous extension

#continuous extension #removable singularity

Ex.870cosh x ≤ e^(x²/2)

#series comparison #hyperbolic functions #inequalities

Taylor expansions

Ex.871Circular arrangements

#circular permutations #necklace #seating

Counting principles

Ex.872Arrangements with repetition

#permutations with repetition #multinomial

Counting principles

Ex.873A bijective proof

#bijection #combinatorial proof #identity

Counting principles

Ex.874Ordering books on a shelf

#permutations #block arrangement #product

Counting principles

Ex.875Counting Boolean functions

#Boolean functions #counting #product rule

Counting principles

Ex.876Counting cliques

#cliques #graphs #counting

Counting principles

Ex.877Coloring problems

#coloring #product rule #recurrence

Counting principles

Ex.878Combinations with repetition

#combinations with repetition #stars and bars

Counting principles

Ex.879Complement counting

#complement counting #inclusion-exclusion #committee

Counting principles

Ex.880Degrees in a graph

#graph degree #handshake lemma #parity

Counting principles

Ex.881Handshake lemma

#double counting #graph #degree sum

Counting principles

Ex.882Counting pairs within subsets

#double counting #subsets #pairs

Counting principles

Ex.883Counting with at least one

#inclusion-exclusion #at least one #complement

Counting principles

Ex.884At least one property

#inclusion-exclusion #properties #at least one

Counting principles

Ex.885Counting unions

#inclusion-exclusion #union of sets #counting

Counting principles

Ex.886Counting configurations

#inclusion-exclusion #configurations #chessboard

Counting principles

Ex.887Sets of even size

#even subsets #binomial #counting

Counting principles

Ex.888Exact intersection count

#inclusion-exclusion #exact count #sets

Counting principles

Ex.889Counting onto functions

#onto #surjections #inclusion-exclusion

Counting principles

Ex.890Monotone lattice paths

#lattice paths #binomial #counting

Counting principles

Ex.891Pigeonhole with integers

#pigeonhole #consecutive integers #differences

Counting principles

Ex.892Generalized pigeonhole

#pigeonhole #generalized #ceil

Counting principles

Ex.893Pigeonhole proof

#pigeonhole #proof #contradiction

Counting principles

Ex.894Subset sums

#subset sums #double counting #average

Counting principles

Ex.895Scheduling tournaments

#scheduling #bipartite #matchings

Counting principles

Ex.896The determinant is alternating

#alternating #equal columns #sign change

Ex.897Determinant of a block triangular matrix

#block matrices #determinant #triangular

Ex.898A trigonometric determinant

#trigonometric determinant #cos #sin #identity

Ex.899Cramer's rule for a 2×2 system

#Cramer's rule #linear system #determinant

Ex.900Determinant of a cyclic permutation matrix

#permutation matrix #cyclic #determinant

Ex.901Determinant of a diagonalisable matrix

#diagonalisation #determinant #eigenvalues

Ex.902Expansion along a column

#cofactor expansion #minors #determinant

Ex.903Expansion along a row

#cofactor expansion #minors #determinant

Ex.904Multilinearity of the determinant

#multilinearity #determinant properties #proof

Ex.905Determinant of an orthogonal matrix

#orthogonal matrix #determinant #isometry

Ex.906Determinant and rank

#determinant #rank #invertibility

Ex.907Determinant of the transpose

#transpose #determinant #proof

Ex.908Bézout's identity

#Bézout #Euclidean algorithm #linear combination

Divisibility and gcd

Ex.909Coprime product

#coprime #product #divisibility

Divisibility and gcd

Ex.910Infinitely many primes

#Euclid #primes #proof by contradiction

Divisibility and gcd

Ex.911The fundamental theorem of arithmetic

#fundamental theorem #prime factorization #uniqueness

Divisibility and gcd

Ex.912Gauss's lemma

#Gauss lemma #divisibility #coprime

Divisibility and gcd

Ex.913GCD and product of coprime factors

#gcd #coprime #factorization

Divisibility and gcd

Ex.914Divisors of coprime numbers

#coprime #divisors #factorization

Divisibility and gcd

Ex.915Linear relations and coprimality

#coprime #linear relation #proof

Divisibility and gcd

Ex.916Proof of the Euclidean algorithm

#Euclidean algorithm #gcd invariance #proof

Divisibility and gcd

Ex.917GCD as a linear combination

#gcd #linear combination #Bézout

Divisibility and gcd

Ex.918GCD and linear combinations

#gcd #linear combination #divisors

Divisibility and gcd

Ex.919GCD of powers

#gcd #powers #prime factorization

Divisibility and gcd

Ex.920A coprimality proof

#coprime #proof #linear combination

Divisibility and gcd

Ex.921A recursive GCD

#gcd #Fibonacci #recursion

Divisibility and gcd

Ex.922Semiprimes and divisors

#semiprime #divisors #factorization

Divisibility and gcd

Ex.923Additive property of the p-adic valuation

#p-adic valuation #additivity #non-Archimedean

Divisibility and gcd

Ex.924Finding modular inverses

#modular inverse #Bézout #congruences

Divisibility and gcd

Ex.925Number of divisors

#divisors #prime factorization #counting

Divisibility and gcd

Ex.926Perfect numbers

#perfect numbers #sum of divisors #Euclid

Divisibility and gcd

Ex.927Perfect squares and divisors

#perfect squares #divisors #parity

Divisibility and gcd

Ex.928Prime power divisibility

#prime power #valuation #p-adic

Divisibility and gcd

Ex.929The radical of an integer

#radical #squarefree #prime factorization

Divisibility and gcd

Ex.930Sum of divisors

#sum of divisors #multiplicative #formula

Divisibility and gcd

Ex.931Units modulo $n$

#units #invertible #Euler totient

Divisibility and gcd

Ex.932p-adic valuation

#p-adic valuation #divisibility #properties

Divisibility and gcd

Ex.933A 2×2 matrix with a double eigenvalue

#eigenvalues #double root #diagonalizability

Eigenvalues and diagonalisation

Ex.934A 3×3 matrix with distinct eigenvalues

#eigenvalues #eigenvectors #3x3 matrices

Eigenvalues and diagonalisation

Ex.935Eigenvalues of a symmetric matrix

#symmetric matrix #eigenvalues #trace and determinant

Eigenvalues and diagonalisation

Ex.936The characteristic polynomial

#characteristic polynomial #eigenvalues #trace #determinant

Eigenvalues and diagonalisation

Ex.937Eigenvalues of a rotation

#rotation #complex eigenvalues #2x2 matrices

Eigenvalues and diagonalisation

Ex.938Similar matrices have the same eigenvalues

#similarity #eigenvalues #characteristic polynomial

Eigenvalues and diagonalisation

Ex.939Powers via diagonalization

#diagonalization #powers #matrix

Eigenvalues and diagonalisation

Ex.940A basis of an eigenspace

#eigenspace #basis #dimension

Eigenvalues and diagonalisation

Ex.941Sum of eigenvalues equals trace

#eigenvalues #trace #characteristic polynomial

Eigenvalues and diagonalisation

Ex.942Trace and determinant of a 3×3 matrix via eigenvalues

#trace #determinant #eigenvalues

Eigenvalues and diagonalisation

Ex.943A biquadratic equation

#biquadratic #substitution #real roots

Equations and radicals

Ex.944A cubic with three roots

#cubic #factoring #roots

Equations and radicals

Ex.945A cubic by factoring

#cubic #factoring #rational root

Equations and radicals

Ex.946Vieta's formulas

#Vieta #sum and product of roots #quadratic

Equations and radicals

Ex.947A quartic by factoring

#quartic #factoring #roots

Equations and radicals

Ex.948A cube root equation

#cube root #cubing #real solutions

Equations and radicals

Ex.949A radical inequality

#radical inequality #squaring #cases

Equations and radicals

Ex.950A radical equation with two square roots

#radical #two roots #conditions

Equations and radicals

Ex.951A radical equation by substitution

#radical #substitution #quadratic

Equations and radicals

Ex.952Two square roots

#radical equation #squaring twice #conditions

Equations and radicals

Ex.953A 3×3 linear system

#linear system #3 equations #elimination

Equations and radicals

Ex.954A nonlinear system

#nonlinear system #elimination #symmetry

Equations and radicals

Ex.955A symmetric system

#symmetric system #substitution #Vieta

Equations and radicals

Ex.956Finite difference formulas for f′

#finite differences #numerical differentiation #error estimate

Taylor expansions

Ex.957BFS and DFS

#BFS #DFS #traversal

Ex.958Bipartite graphs

#bipartite #2-coloring #odd cycle

Ex.959Cliques and independent sets

#clique #independent set #complement

Ex.960A graph with a unique degree sequence

#degree sequence #unique #graph

Ex.961Hamiltonian cycles (example)

#Hamiltonian #cycle #NP-hard

Ex.962Graph isomorphism

#isomorphism #bijection #invariant

Ex.963Matchings

#matching #bipartite #Hall

Ex.964Euler's formula application

#Euler formula #planar #faces

Ex.965Spanning trees

#spanning tree #subgraph #connectivity

Ex.966Characterizations of trees

#tree #characterization #connected

Ex.967Diameter of a tree

#tree #diameter #BFS

Ex.968Trees as minors

#tree #minor #contraction

Ex.969The center of a group

#center #abelian #subgroup

Ex.970Counting cosets

#cosets #index #counting

Ex.971Groups of prime order

#prime order #cyclic #subgroups

Ex.972The general linear group $GL_n(\mathbb{R})$

#general linear group #matrices #invertibility

Ex.973Group homomorphisms

#homomorphism #kernel #image

Ex.974Homomorphisms between familiar groups

#homomorphism #examples #kernels

Ex.975Subgroups of index 2 are normal

#index #normal #cosets

Ex.976Isomorphism criteria

#isomorphism #bijective homomorphism #criterion

Ex.977Injectivity and kernel

#kernel #injectivity #homomorphism

Ex.978Corollary of Lagrange's theorem

#Lagrange #order of element #corollary

Ex.979Lagrange's theorem

#Lagrange #order #subgroup

Ex.980Order of an element in $\mathbb{Z}/n\mathbb{Z}$

#order #cyclic group #gcd

Ex.981Direct products of groups

#direct product #product group #order

Ex.982Subgroup lattice of $\mathbb{Z}/6\mathbb{Z}$

#subgroup lattice #cyclic group #divisors

Ex.983The symmetric group $S_3$

#symmetric group #permutations #order

Ex.984Transpositions generate $S_n$

#transpositions #generation #symmetric group

Ex.985Derivatives of high order at 0 without differentiating

#Taylor expansion #uniqueness #high order derivatives

Taylor expansions

Ex.986Permutations with at least one fixed point

#IE #fixed point #permutations

Inclusion-exclusion

Ex.987Binomial identity via IE

#IE #binomial #identity

Inclusion-exclusion

Ex.988Coprime to $100$

#IE #coprime #totient

Inclusion-exclusion

Ex.989Numbers coprime to $30$ up to $100$

#IE #coprime #totient

Inclusion-exclusion

Ex.990Counting numbers with at least one property

#IE #counting #divisibility

Inclusion-exclusion

Ex.991Derangements of $5$ elements

#derangements #$D_5$ #computation

Inclusion-exclusion

Ex.992Derangements of $6$ elements

#derangements #$D_6$ #computation

Inclusion-exclusion

Ex.993Derangement recurrence

#derangement #recurrence #IE

Inclusion-exclusion

Ex.994Derangements

#derangements #permutations #no fixed point

Inclusion-exclusion

Ex.995Euler's totient via inclusion-exclusion

#Euler totient #IE #coprime

Inclusion-exclusion

Ex.996Exactly one property

#IE #exactly one #properties

Inclusion-exclusion

Ex.997The general inclusion-exclusion formula

#inclusion-exclusion #general #n sets

Inclusion-exclusion

Ex.998Items with no defect

#IE #complement #no defect

Inclusion-exclusion

Ex.999Letters and envelopes

#IE #derangements #letters

Inclusion-exclusion

Ex.1000Elements with none of the properties

#IE #complement #none

Inclusion-exclusion

Ex.1001Surjections from $5$ to $3$

#surjections #IE #count

Inclusion-exclusion

Ex.1002Inclusion-exclusion in a survey

#IE #survey #population

Inclusion-exclusion

Ex.1003Counting with divisibility

#IE #divisibility #counting

Inclusion-exclusion

Ex.1004The sieve of Eratosthenes as IE

#IE #sieve #primes

Inclusion-exclusion

Ex.1005Surjections from $4$ to $3$

#surjections #IE #count

Inclusion-exclusion

Ex.1006Counting surjections via inclusion-exclusion

#surjections #onto #IE

Inclusion-exclusion

Ex.1007Partial three-set data

#IE #three sets #intersection

Inclusion-exclusion

Ex.1008Exactly one of two properties

#IE #exactly one #properties

Inclusion-exclusion

Ex.1009Two triples

#IE #triples #sets

Inclusion-exclusion

Ex.1010IE with complements

#IE #complements #sets

Inclusion-exclusion

Ex.1011AM-GM for three numbers

#AM-GM #three variables #inequality

Ex.1012The arithmetic-harmonic mean inequality

#AM-HM #harmonic mean #inequality

Ex.1013Bernoulli's inequality

#Bernoulli #inequality #induction

Ex.1014Cauchy-Schwarz by induction

#Cauchy-Schwarz #induction #inequality

Ex.1015Cauchy-Schwarz inequality

#Cauchy-Schwarz #inequality #sequences

Ex.1016Geometric mean and harmonic mean

#geometric mean #harmonic mean #inequality

Ex.1017Jensen's inequality for the square function

#Jensen #convexity #square function

Ex.1018Discriminant and positivity

#discriminant #positivity #quadratic

Ex.1019The rearrangement inequality

#rearrangement #inequality #sequences

Ex.1020Titu's lemma (Cauchy in Engel form)

#Titu #Engel form #Cauchy-Schwarz

Ex.1021Gram-Schmidt orthonormalization

#Gram-Schmidt #orthonormal basis #algorithm

Inner product spaces

Ex.1022An orthogonal basis from the standard basis

#orthogonal basis #Gram-Schmidt #polynomial space

Inner product spaces

Ex.1023Orthogonal complement

#orthogonal complement #dimension #direct sum

Inner product spaces

Ex.1024Orthogonal matrices and isometries

#orthogonal matrix #isometry #inner product preservation

Inner product spaces

Ex.1025Triangle inequality

#triangle inequality #norm #inner product

Inner product spaces

Ex.1026Rational root theorem and irrationality

#rational root theorem #algebraic #irrationality

Ex.1027Uncountability of irrationals

#uncountability #Cantor #irrationals

Ex.1028Rational coefficients and irrational roots

#rational coefficients #irrational roots #conjugates

Ex.1029Degree of algebraic numbers

#algebraic degree #minimal polynomial #sqrt2

Ex.1030Rational, algebraic, transcendental

#hierarchy #algebraic #transcendental

Ex.1031Proving an equivalence

#equivalence #iff #proof

Proof techniques

Ex.1032Coins and strong induction

#induction #coins #Frobenius

Ex.1033Full binary tree node count

#induction #binary tree #structural

Ex.1034Divisibility by induction

#induction #divisibility #powers

Ex.1035An identity for Fibonacci numbers

#induction #Fibonacci #identity

Ex.1036Towers of Hanoi

#induction #Hanoi #recurrence

Ex.1037Exponential inequality by induction

#induction #inequality #exponential

Ex.1038Fibonacci growth bound

#induction #Fibonacci #growth

Ex.1039$n!$ vs $2^n$

#induction #factorial #exponential

Ex.1040Induction and periodicity

#induction #periodicity #recurrence

Ex.1041Product by induction

#induction #product #telescoping

Ex.1042Strong induction

#strong induction #prime factorization

Ex.1043Sum of cubes

#induction #sum of cubes

Ex.1044A factorial identity

#induction #factorial #identity

Ex.1045Existence proofs

#existence #constructive #non-constructive

Proof techniques

Ex.1046Order of quantifiers

#quantifiers #order #logic

Proof techniques

Ex.1047Approximating x by a combination of sines

#undetermined coefficients #Taylor expansion #Richardson extrapolation

Taylor expansions

Ex.1048Block multiplication

#block matrices #multiplication #structure

Matrices and linear systems

Ex.1049Change of basis

#change of basis #transition matrix #similarity

Matrices and linear systems

Ex.1050Cramer's rule

#Cramer's rule #determinant #linear system

Matrices and linear systems

Ex.1051Determinant of a block triangular matrix

#block matrices #determinant #triangular structure

Matrices and linear systems

Ex.1052Using determinant properties

#determinant #row operations #properties

Matrices and linear systems

Ex.1053Eigenvalues and eigenvectors of a 2×2 matrix

#eigenvalues #eigenvectors #characteristic polynomial

Matrices and linear systems

Ex.1054Solving a linear system by Gauss-Jordan

#linear system #Gauss-Jordan #row reduction

Matrices and linear systems

Ex.1055Gram-Schmidt orthogonalization

#Gram-Schmidt #orthogonalization #orthonormal basis

Matrices and linear systems

Ex.1056Invertibility depending on a parameter

#invertibility #parameter #determinant

Matrices and linear systems

Ex.1057Kernel and image of a matrix

#kernel #image #basis

Matrices and linear systems

Ex.1058Rank-nullity theorem

#rank-nullity #kernel #image

Matrices and linear systems

Ex.1059Powers of a nilpotent matrix

#nilpotent #powers #induction

Matrices and linear systems

Ex.1060Computing the rank of a matrix

#rank #Gaussian elimination #row reduction

Matrices and linear systems

Ex.1061Similarity invariants

#similarity #characteristic polynomial #trace #determinant

Matrices and linear systems

Ex.1062Compactness in metric spaces

#compactness #open cover #sequential

Metric and normed spaces

Ex.1063Compact sets are closed and bounded

#compact #closed #bounded #metric

Metric and normed spaces

Ex.1064Continuous image of compact

#compact image #continuity #compactness

Metric and normed spaces

Ex.1065Complete metric spaces

#completeness #Cauchy sequences #examples

Metric and normed spaces

Ex.1066Equivalent metrics

#equivalent metrics #same topology #norms

Metric and normed spaces

Ex.1067Isometries between metric spaces

#isometry #metric #distance-preserving

Metric and normed spaces

Ex.1068Equivalent norms

#equivalent norms #topology #finite dimension

Metric and normed spaces

Ex.1069Classical normed spaces

#normed spaces #Banach #examples

Metric and normed spaces

Ex.1070Product metrics

#product spaces #metric #sup

Metric and normed spaces

Ex.1071Boundary effects

#boundary #integration bounds #evaluation

Multiple integrals

Ex.1072Cartesian vs polar computation

#polar vs cartesian #comparison #disk

Multiple integrals

Ex.1073Double integral over an annulus

#polar coordinates #annulus #angular integral

Multiple integrals

Ex.1074Double integral over a disk tangent at origin

#polar coordinates #circle offset #integral

Multiple integrals

Ex.1075Double integral over a disk in polar coordinates

#polar coordinates #disk #Jacobian

Multiple integrals

Ex.1076Polar integral over a half-disk

#polar #half-disk #integral

Multiple integrals

Ex.1077Mass of a cube with variable density

#triple integral #density #cube

Multiple integrals

Ex.1078Triple integral over a simplex

#triple integral #simplex #iterated

Multiple integrals

Ex.1079General chain rule

#chain rule #composition #multivariable

Partial derivatives

Ex.1080Clairaut's theorem

#Clairaut #Schwarz #mixed partials

Partial derivatives

Ex.1081Saddle point

#saddle point #Hessian #classification

Optimisation in several variables

Ex.1082The differential

#differential #linearization #approximation

Partial derivatives

Ex.1083Second-order partial derivatives

#higher order #Schwarz #second derivatives

Partial derivatives

Ex.1084The Laplacian

#Laplacian #harmonic functions #second derivatives

Partial derivatives

Ex.1085Finding critical points

#critical points #stationary #gradient

Partial derivatives

Ex.1086Schwarz's theorem and a counterexample

#Schwarz theorem #mixed partials #symmetry

Partial derivatives

Ex.1087Tangent plane to a graph

#tangent plane #differentiability #graph

Partial derivatives

Ex.1088$\{1,2,3,\ldots\}$ and pairwise differences

#pigeonhole #difference #arithmetic

Pigeonhole principle

Ex.1089Eratosthenes via pigeonhole

#pigeonhole #primes #sieve

Pigeonhole principle

Ex.1090Sum of digits

#pigeonhole #digits #sums

Pigeonhole principle

Ex.1091Binomial coefficient bound

#pigeonhole #binomial #subset size

Pigeonhole principle

Ex.1092Consecutive integers whose sum is divisible

#pigeonhole #consecutive sum #divisible

Pigeonhole principle

Ex.1093Divisibility via pigeonhole

#pigeonhole #divisibility #subsets

Pigeonhole principle

Ex.1094Three same-color integers

#pigeonhole #same color #arithmetic progression

Pigeonhole principle

Ex.1095Points in an equilateral triangle

#pigeonhole #equilateral #distance

Pigeonhole principle

Ex.1096An even number of odd vertices

#pigeonhole #graph #parity

Pigeonhole principle

Ex.1097The handshake lemma by pigeonhole

#pigeonhole #handshake #graph degrees

Pigeonhole principle

Ex.1098Integer grid points

#pigeonhole #grid #midpoints

Pigeonhole principle

Ex.1099Pigeonhole in a Latin square

#pigeonhole #Latin square #entries

Pigeonhole principle

Ex.1100Many odd numbers

#pigeonhole #odd #sums

Pigeonhole principle

Ex.1101Modular pigeonhole

#pigeonhole #modular #addition

Pigeonhole principle

Ex.1102A multiple of $n$ from $0$s and $1$s

#pigeonhole #multiples #digits

Pigeonhole principle

Ex.1103An odd number in a triple

#pigeonhole #parity #odd

Pigeonhole principle

Ex.1104Points in a unit square

#pigeonhole #geometry #distance

Pigeonhole principle

Ex.1105Subset sums

#pigeonhole #subset sums #olympiad

Pigeonhole principle

Ex.1106Remainders of consecutive integers

#pigeonhole #remainders #consecutive

Pigeonhole principle

Ex.1107Pigeonhole in a table

#pigeonhole #table #rows

Pigeonhole principle

Ex.1108A sequence of $0$s and $1$s

#pigeonhole #sequence #blocks

Pigeonhole principle

Ex.1109Composition of polynomials

#composition #polynomials #degree

Ex.1110The discriminant of a quadratic

#discriminant #quadratic #roots

Ex.1111Divisibility of polynomials

#divisibility #roots #factorisation

Ex.1112Factorization of a cubic

#cubic #factorization #roots

Ex.1113Multiplicity of a root

#multiplicity #repeated root #derivative

Ex.1114The rational root test

#rational root theorem #factoring #roots

Ex.1115Factorization over $\mathbb{R}$

#factorization #real roots #complex roots

Ex.1116Sum of roots of unity

#roots of unity #sum #cyclotomic

Ex.1117Vieta's formulas for a cubic

#Vieta #cubic #symmetric functions

Ex.1118Coprimality and Euler's theorem

#coprime #Euler theorem #application

Ex.1119Coprimality with factorials

#coprime #factorial #prime

Ex.1120Factorial and coprime integers

#factorial #coprime #bounds

Ex.1121Consecutive primes and products

#consecutive primes #product #gap

Ex.1122Coprime integers and prime divisors

#coprime #prime divisors #intersection

Ex.1123Coprime products and primes

#coprime #product #prime

Ex.1124Divisibility by coprime integers

#coprime #divisibility #product

Ex.1125Coprime remainders

#coprime #remainders #CRT

Ex.1126Multiplicativity of $\varphi$

#Euler totient #multiplicative #CRT

Ex.1127Euler's totient of a prime power

#Euler totient #prime power #formula

Ex.1128Factoring a large integer

#factorization #primes #algorithm

Ex.1129Fermat's little theorem

#Fermat #prime #modular exponentiation

Ex.1130Fermat primality test

#Fermat test #primality #witness

Ex.1131Fermat's little theorem and compositeness

#Fermat #compositeness test #modular exponentiation

Ex.1132Totient and coprime residue classes

#totient #coprime #residue classes

Ex.1133Euclid's lemma

#Euclid lemma #prime #divisibility

Ex.1134Primes and factorials

#factorial #primes #Wilson

Ex.1135GCD of a product

#gcd #product #prime factorization

Ex.1136Mersenne primes

#Mersenne #primes #perfect numbers

Ex.1137Mersenne primes and exponents

#Mersenne #primes #exponents

Ex.1138Order modulo a prime

#order #prime #divisor

Ex.1139Even perfect numbers and Mersenne primes

#perfect numbers #Mersenne #Euclid-Euler

Ex.1140The prime counting function

#prime counting #pi function #estimates

Ex.1141Prime gaps

#prime gaps #arbitrary gaps #construction

Ex.1142Prime powers and divisibility

#prime powers #divisibility #exponents

Ex.1143Primality test up to $\sqrt n$

#primality test #sqrt #trial division

Ex.1144Sophie Germain primes

#Sophie Germain #primes #examples

Ex.1145Reducing a quadratic form by completing the square

#reduction #completing square #quadratic form

Quadratic forms

Ex.1146Negative definite quadratic form

#negative definite #signature #Sylvester

Quadratic forms

Ex.1147A degenerate quadratic form

#degenerate #signature #quadratic form

Quadratic forms

Ex.1148The hyperbola $x^2-y^2=1$

#hyperbola #quadratic form #geometry

Quadratic forms

Ex.1149An indefinite quadratic form

#indefinite #signature #quadratic form

Quadratic forms

Ex.1150Quantifier absorption

#quantifier absorption #vacuous quantifier

Quantifiers and negation

Ex.1151Renaming bound variables

#alpha-conversion #bound variable #renaming

Quantifiers and negation

Ex.1152De Morgan's laws for quantifiers

#De Morgan #quantifiers #negation

Quantifiers and negation

Ex.1153The $\varepsilon$-$\delta$ definition of limit

#epsilon-delta #quantifiers #limit

Quantifiers and negation

Ex.1154Existence and uniqueness proof

#existence #uniqueness #proof

Quantifiers and negation

Ex.1155A false $\exists\forall$ statement

#exists forall #false #counterexample

Quantifiers and negation

Ex.1156A quantified implication

#forall implies #existential #mixed

Quantifiers and negation

Ex.1157Free and bound variables

#free variable #bound variable #quantifier

Quantifiers and negation

Ex.1158Quantifiers in induction proofs

#induction #quantifiers #base case

Quantifiers and negation

Ex.1159Quantifiers in convergence of sequences

#convergence #epsilon #sequences

Quantifiers and negation

Ex.1160Negating nested quantifiers

#negation #nested quantifiers #forall exists

Quantifiers and negation

Ex.1161Negating the uniqueness quantifier

#negation #uniqueness #exists unique

Quantifiers and negation

Ex.1162Order of quantifiers matters

#quantifier order #forall exists #difference

Quantifiers and negation

Ex.1163Vacuous truth

#vacuous truth #forall #empty set

Quantifiers and negation

Ex.1164Hasse diagram

#Hasse diagram #covering #poset

Ex.1165Equivalence classes form a partition

#partition #equivalence classes #proof

Ex.1166Partitions induce equivalence relations

#partition #equivalence #converse

Ex.1167Total orders

#total order #comparability #real numbers

Ex.1168Characteristic of a ring

#characteristic #ring #examples

Rings and fields

Ex.1169Euclidean division in $\mathbb{R}[X]$

#Euclidean division #polynomials #quotient and remainder

Rings and fields

#ideal #definition #examples

Rings and fields

Ex.1171Ideals in $\mathbb{Z}$

#ideals #Z #principal ideal

Rings and fields

Ex.1172Matrix rings and zero divisors

#matrix ring #zero divisors #non-commutative

Rings and fields

Ex.1173The polynomial ring $R[X]$

#polynomial ring #properties #integral domain

Rings and fields

Ex.1174Prime ideals in $\mathbb{Z}$

#prime ideals #Z #primes

Rings and fields

Ex.1175Convergence of three series via expansions

#series #equivalents #comparison

Taylor expansions

Ex.1176Arbitrary unions and intersections

#arbitrary union #arbitrary intersection #family

Ex.1177Equivalence classes and partitions

#equivalence class #partition #quotient

Ex.1178Equivalence relations

#equivalence relation #reflexive #symmetric #transitive

Ex.1179Partial orders

#partial order #poset #relation

Ex.1180Total orders

#total order #comparability #poset

Ex.1181Expanding x/(eˣ − 1)

#Taylor expansion #quotient #Bernoulli numbers #undetermined coefficients

Taylor expansions

Ex.1182Taylor series of the solution of y′ = x + y²

#differential equation #Taylor expansion #successive derivatives

Taylor expansions

Ex.1183The symmetric second difference

#Taylor-Young #second derivative #counterexample

Taylor expansions

Ex.1184Interior, closure, boundary

#interior #closure #boundary

Topology of ℝ

Ex.1185Connected subsets of $\mathbb{R}$

#connectedness #intervals #IVT

Topology of ℝ

Ex.1186Limit points

#limit points #accumulation #closure

Topology of ℝ

Ex.1187Dense sets

#dense #rationals #irrationals

Topology of ℝ

Ex.1188Heine–Borel in $\mathbb{R}^n$

#compactness #Heine-Borel #closed bounded

Topology of ℝⁿ

Ex.1189Continuous image of a compact set

#compact #continuous image #extrema

Topology of ℝⁿ

Ex.1190Connectedness in $\mathbb{R}^n$

#connectedness #path-connectedness

Topology of ℝⁿ

Ex.1191Convex sets are connected

#connectedness #convexity #segments

Topology of ℝⁿ

Ex.1192Continuity versus iterated limits

#iterated limits #continuity #subtleties

Topology of ℝⁿ

Ex.1193Sequential compactness in $\mathbb{R}^n$

#Bolzano-Weierstrass #sequential compactness

Topology of ℝⁿ

Ex.1194A basis of a hyperplane in $\mathbb{R}^4$

#basis #hyperplane #dimension

Vector spaces and subspaces

Ex.1195Image of a basis under a linear map

#basis #image #linear map

Vector spaces and subspaces

Ex.1196A basis of a polynomial space

#polynomial spaces #basis #dimension

Vector spaces and subspaces

Ex.1197The canonical basis and matrix representation

#canonical basis #matrix representation #linear map

Vector spaces and subspaces

Ex.1198The curl

#curl #vector field #rotation

Vector calculus

Ex.1199Eigenspaces as subspaces

#eigenspaces #subspaces #dimension

Vector spaces and subspaces

Ex.1200Dimension of a subspace

#dimension #subspace #equations

Vector spaces and subspaces

Ex.1201Flux through a flat surface

#flux #plane surface #dot product

Vector calculus

Ex.1202An application of Green's theorem

#Green #application #area

Vector calculus

Ex.1203Area using Green's theorem

#area #Green #closed curve

Vector calculus

Ex.1204Green with symmetry

#Green #symmetry #shortcut

Vector calculus

Ex.1205Independence depending on a parameter

#linear independence #parameter #determinant

Vector spaces and subspaces

Ex.1206Isomorphism criterion

#isomorphism #dimension #injective linear map

Vector spaces and subspaces

Ex.1207Basis of a kernel

#kernel #basis #linear map

Vector spaces and subspaces

Ex.1208Kernel and image of a linear map

#kernel #image #linear map #rank-nullity

Vector spaces and subspaces

Ex.1209Kernel of a differentiation-like map

#kernel #differentiation #polynomials

Vector spaces and subspaces

Ex.1210Conservative vector field

#conservative #gradient #path independence

Vector calculus

Ex.1211Fundamental theorem for line integrals

#fundamental theorem #line integral #potential

Vector calculus

Ex.1212Line integral along a circle

#line integral #circle #parametrization

Vector calculus

Ex.1213Line integral on a piecewise path

#piecewise path #line integral #work

Vector calculus

Ex.1214Work done by a force field

#work #line integral #vector field

Vector calculus

Ex.1215Testing whether a field is conservative

#conservative #curl #test

Vector calculus

Ex.1216The space of linear maps

#linear maps #vector space structure #dimension

Vector spaces and subspaces

Ex.1217Injective linear map and kernel

#injectivity #kernel #linear map

Vector spaces and subspaces

Ex.1218Column space and row space

#column space #row space #rank

Vector spaces and subspaces

Ex.1219Kernel of a 3×3 matrix and basis

#kernel #basis #matrix #rank-nullity

Vector spaces and subspaces

Ex.1220Subspace generated by a family

#span #generated subspace #dimension

Vector spaces and subspaces

Ex.1221Subspace generated by columns of a matrix

#column space #generators #rank

Vector spaces and subspaces

Ex.1222Subspace inclusion and dimension

#inclusion #dimension #equality

Vector spaces and subspaces

Ex.1223The direct sum criterion

#direct sum #criterion #proof

Vector spaces and subspaces

Ex.1224Surjective linear map and image

#surjectivity #image #linear map

Vector spaces and subspaces

Ex.1225The integer part of a telescoping sum

#recurrence #telescoping #floor #monotone sequence

Equations and radicals

Ex.1226cos(π/7) − cos(2π/7) + cos(3π/7) = 1/2

#trigonometry #telescoping #product-to-sum

Equations and radicals

Ex.1227The sum of 1/(1 − ω^k)

#roots of unity #symmetry #complex numbers #pairing

Complex numbers

Ex.1228A recurrence with a hidden invariant

#linear recurrence #invariant #characteristic equation

Equations and radicals

Ex.1229Three unit complex numbers with zero sum

#complex numbers #unit circle #equilateral triangle #conjugate trick

Complex numbers

Ex.1230Hermite's identity for floor functions

#floor function #fractional part #Hermite identity

Equations and radicals

Ex.1231Divisibility of values of an integer polynomial

#integer polynomials #a−b divides P(a)−P(b) #integer roots

Ex.1232Two symmetric systems

#symmetric system #Vieta #sums of squares

Equations and radicals

Ex.1233Telescoping with arctangents and products

#telescoping #arctangent #products

Equations and radicals

Ex.1234The game of coins on a round table

#games #symmetry #winning strategy

Counting principles

Ex.1235Derangements and the number 1/e

#inclusion-exclusion #permutations #derangements

Inclusion-exclusion

Ex.1236Every tournament has a Hamiltonian path

#induction #tournaments #graphs #Hamiltonian path

Ex.1237Finding the heavy coin with a balance

#algorithms #information #induction #weighings

Counting principles

Ex.1238Flipping rows and columns of a board

#invariant #parity #colouring #moves

Counting principles

Ex.1239A tree has at least as many leaves as its maximum degree

#trees #graphs #handshake lemma #degrees

Ex.1240Monotone subsequences (Erdős–Szekeres)

#pigeonhole #sequences #Erdős–Szekeres

Pigeonhole principle

Ex.1241A finite set in which every point is a midpoint

#extremal principle #plane geometry #midpoints

Counting principles

Ex.1242Choosing n + 1 numbers among 1, …, 2n

#pigeonhole #divisibility #odd part

Pigeonhole principle

Ex.1243Subsets without two consecutive integers

#counting #recurrences #Fibonacci #bijection

Counting principles

Ex.1244Tiling a 10 × 10 board with 1 × 4 tiles

#colouring #tilings #tetrominoes #modular colouring

Counting principles

Ex.1245Splitting a parliament into two houses

#monovariant #graphs #extremal #process

Ex.1246Tiling a 2ⁿ × 2ⁿ board with L-trominoes

#induction #tilings #trominoes #divide and conquer

Counting principles

Ex.1247Binary strings with no three equal consecutive digits

#recurrences #Fibonacci numbers #compositions #binary strings

Counting principles

Ex.1248A graph is bipartite exactly when it has no odd cycle

#graph theory #bipartite graphs #colouring #cycles

Ex.1249When can a graph be drawn without lifting the pen?

#Euler circuit #graph theory #degrees #extremal trail

Ex.1250Gray codes: a cyclic list of all bit strings with one change at a time

#Gray code #induction #hypercube #constructions

Counting principles

Ex.1251The fewest kings controlling the board

#covering #extremal #chessboard #lower bound

Pigeonhole principle

Ex.1252Knights and liars around a table

#logic puzzles #circular arrangements #constructions

Pigeonhole principle

Ex.1253The expected number of records

#expectation #symmetry #harmonic numbers #permutations

Counting principles

Ex.1254Sorting by adjacent swaps needs exactly the number of inversions

#permutations #inversions #sorting #monovariant

Counting principles

Ex.1255Stars and bars with upper bounds

#stars and bars #inclusion-exclusion #counting #compositions

Inclusion-exclusion

Ex.1256Surjections and splitting a set into three groups

#Stirling numbers #inclusion-exclusion #surjections #set partitions

Inclusion-exclusion

Ex.1257Gas stations on a circular road

#cyclic sequences #prefix sums #extremal choice #cycle lemma

Counting principles

Ex.1258f(x + f(y)) = f(x) + y on the rationals

#injectivity #involution #Cauchy equation on Q

Functional equations

Ex.1259The Gergonne point and Ceva's theorem

#Ceva's theorem #incircle #tangent lengths

Ex.1260Ptolemy and a point on the circumcircle of an equilateral triangle

#Ptolemy's theorem #cyclic quadrilateral #equilateral triangle

Ex.1261The area of a right triangle from the incircle touch point

#incircle #tangent lengths #Pythagoras #right triangle

Ex.1262The radical axis bisects the common tangent

#power of a point #radical axis #tangent

Ex.1263The reflection of the orthocentre lies on the circumcircle

#angle chasing #cyclic quadrilateral #orthocentre

Ex.1264Two circles, two secants and a pair of parallel lines

#directed angles #cyclic quadrilateral #parallel lines

Ex.1265Chebyshev's sum inequality and a^a b^b c^c

#Chebyshev's sum inequality #logarithm #monotone sequences

Ex.1266A sum of squares of a + 1/a, via Jensen

#Jensen's inequality #convexity #constraint a+b+c=1

Ex.1267The tangent line trick

#tangent line trick #constraint a+b+c=1 #one-variable reduction

Ex.1268An arctangent inequality

#mean value theorem #inequality #arctan

Mean Value Theorem

Ex.1269An exponential inequality

#mean value theorem #inequality #exponential

Mean Value Theorem

Ex.1270A logarithm inequality via the MVT

#mean value theorem #inequality #logarithm

Mean Value Theorem

Ex.1271Bounding $f(b)-f(a)$

#MVT #bound #interval

Mean Value Theorem

Ex.1272Small-increment estimate

#MVT #small increments #estimate

Mean Value Theorem

Ex.1273Bounded derivative implies Lipschitz

#Lipschitz #bounded derivative #MVT

Mean Value Theorem

Ex.1274Rolle and the sign of $f'$

#Rolle #sign change #derivative

Mean Value Theorem

Ex.1275Rolle on a polynomial with three roots

#Rolle #multiple roots #derivative

Mean Value Theorem

Ex.1276Uniqueness of a root

#Rolle #uniqueness #contrapositive

Mean Value Theorem

Ex.1277Long runs of non-squarefree numbers

#Chinese remainder theorem #squarefree numbers

Divisibility and gcd

Ex.1278Infinitely many primes of the form 4k + 3

#primes #Euclid's argument #residues

Ex.1279n divides 2ⁿ − 1 only for n = 1

#order #Fermat's little theorem #smallest prime divisor

Ex.1280The gcd of 2ᵃ − 1 and 2ᵇ − 1

#gcd #Euclidean algorithm #divisibility

Divisibility and gcd

Ex.128160 divides xyz for every Pythagorean triple

#Pythagorean triples #residues #squares modulo n

Divisibility and gcd

Ex.1282Wilson's theorem and its converse

#Wilson's theorem #modular inverses #primality

Ex.1283Two implications imply the negation of $p$

#implication #truth table #contradiction

Proof techniques

Ex.1284The barber's paradox

#self-reference #paradox #implication

Quantifiers and negation

Ex.1285Three numbers with signs

#sign analysis #implications #logic puzzle

Quantifiers and negation

Ex.1286A non-compact closed subspace: trace-zero matrices

#closed unbounded #trace #non-compactness

Topology of Mn(ℝ)

Ex.1287Completeness of $M_n(\mathbb{R})$

#completeness #Cauchy sequence #identification with R^{n^2}

Topology of Mn(ℝ)

Ex.1288Interior and boundary of $GL_n(\mathbb{R})$

#interior #boundary #GL_n

Topology of Mn(ℝ)

Ex.1289Equivalent metrics on $M_n(\mathbb{R})$

#equivalent metrics #norms on matrices #finite dimension

Topology of Mn(ℝ)

Ex.1290Submultiplicativity of matrix norms

#submultiplicative norms #operator norm #matrix norm

Topology of Mn(ℝ)

Ex.1291Separability of $M_n(\mathbb{R})$

#separability #dense countable subset #rational matrices

Topology of Mn(ℝ)

Ex.1292No polynomial takes only prime values

#integer polynomials #primes #divisibility #congruences

Ex.1293A monic polynomial with root √2 + √3 + √5

#algebraic numbers #minimal polynomial #rational root theorem #irrationality

Ex.1294Three complex numbers of modulus one

#complex numbers #unit circle #conjugation #equilateral triangle

Complex numbers

Ex.1295Every integer divides some Fibonacci number

#Fibonacci numbers #periodicity #pigeonhole #modular arithmetic

Equations and radicals

Ex.1296Finite differences and alternating binomial sums

#finite differences #polynomials #binomial coefficients #degree

Ex.1297How fast does a_{n+1} = a_n + 1/a_n grow?

#sequences #growth #squaring trick #estimates

Equations and radicals

Ex.1298P(x²) = P(x)²

#polynomial equation #coefficients #degree #comparing terms

Ex.1299tan 20° tan 40° tan 80° and tan²20° + tan²40° + tan²80°

#trigonometry #tangent #triple angle #Vieta

Equations and radicals

Ex.1300Approximating √2 by 1 + 1/(1 + x)

#continued fractions #sequences #Pell equation #approximation

Equations and radicals

Ex.1301P′² ≥ P·P″ for polynomials with real roots

#polynomials #real roots #logarithmic derivative #inequalities

Ex.1302The derivative keeps all roots real

#polynomials #Rolle's theorem #multiplicity #real roots

Ex.1303P(x² + 1) = P(x)² + 1

#polynomial equation #recursive sequences #identity theorem

Ex.1304The product cos(π/N)·cos(2π/N)···cos(nπ/N) with N = 2n + 1

#trigonometry #product #double angle #permutation of angles

Equations and radicals

Ex.1305Sums over roots of unity from a logarithmic derivative

#roots of unity #logarithmic derivative #complex numbers #symmetry

Complex numbers

Ex.1306A cut containing at least half of the edges

#probabilistic method #expectation #graph theory #local improvement

Ex.1307A 7 × 7 grid with no rectangle of marked cells

#double counting #convexity #Fano plane #extremal

Counting principles

Ex.1308Vandermonde's identity and a weighted sum of squares

#binomial identities #Vandermonde #double counting

Binomial coefficients

Ex.1309Partitions into distinct parts and into odd parts

#generating functions #partitions #Euler

Counting principles

Ex.1310Drawing K₈ without lifting the pen

#Euler trail #graph theory #degrees #construction

Ex.1311Five points contain a convex quadrilateral

#combinatorial geometry #convex position #pigeonhole #Happy Ending

Pigeonhole principle

Ex.1312Nim and the XOR rule

#combinatorial games #Nim #XOR #binary

Counting principles

Ex.1313The 15-puzzle with two tiles swapped

#invariant #parity #inversions #puzzle

Counting principles

Ex.1314f(x)f(y) − f(xy) = x + y

#functional equation #substitution #polynomial identity

Functional equations

Ex.1315Equilateral triangles and complex numbers

#complex numbers #equilateral triangle #roots of unity #rotation

Analytic geometry

Ex.1316Excircles and tangent lengths

#excircle #incircle #tangent lengths #semiperimeter #area

Ex.1317Pick's theorem and lattice triangles

#Pick's theorem #lattice points #area #equilateral triangle

Analytic geometry

Ex.1318The Apollonius circle

#locus #Apollonius circle #coordinates #angle bisector

Analytic geometry

Ex.1319The distance between the circumcentre and the orthocentre

#vectors #orthocentre #circumcentre #Euler line

Analytic geometry

Ex.1320Two circles and a line through a common point

#spiral similarity #two intersecting circles #chord length #projection

Ex.1321Weitzenböck's inequality

#geometric inequality #Heron's formula #area #sum of squares

Ex.1322A symmetrisation trick: Σ a³/(a² + ab + b²)

#symmetrisation #cyclic sums #tangent-style bound

Ex.1323IMO 1984: yz + zx + xy − 2xyz

#IMO 1984 #pqr method #case analysis #AM-GM

Ex.1324The product of the sines of a triangle

#Jensen #concavity #logarithm #triangle

Ex.1325Minkowski's inequality and a sum of two square roots

#Minkowski #triangle inequality #optimisation

Ex.1326Muirhead's idea with weighted AM-GM

#Muirhead #majorization #weighted AM-GM

Ex.1327Ravi substitution: abc ≥ (a + b − c)(b + c − a)(c + a − b)

#Ravi substitution #triangle inequality #AM-GM

Ex.1328A multiple of n written with only 0s and 1s

#pigeonhole #repunits #decimal digits

Ex.1329(n²)! is divisible by (n!)^(n+1)

#integrality #factorials #double counting #multinomial

Divisibility and gcd

Ex.1330If aᵖ ≡ bᵖ (mod p) then aᵖ ≡ bᵖ (mod p²)

#Fermat's little theorem #binomial theorem #lifting

Ex.1331Infinitely many n dividing 2ⁿ + 1

#induction #lifting the exponent #infinitely many #factorisation

Divisibility and gcd

Ex.1332Infinitely many primes ≡ 1 (mod 4)

#infinitely many primes #order #Euclid's argument

Ex.1333v_p(n!) from the digits of n in base p

#Legendre's formula #digit sum #p-adic valuation #base p

Divisibility and gcd

Ex.1334Counting odd entries in a row of Pascal's triangle

#binomial coefficients #parity #binary expansion #generating function

Ex.1335Prime divisors of 2ᵖ − 1

#Mersenne numbers #order #primality testing

Ex.1336Prime divisors of n² + n + 1

#order #Fermat's little theorem #primes in progressions

Ex.1337((p − 1)/2)! modulo a prime p ≡ 3 (mod 4)

#Wilson's theorem #factorials modulo p #quadratic residues

Ex.1338The Frobenius coin problem for two coins

#Frobenius number #linear Diophantine #Chicken McNugget

Diophantine equations

Ex.1339The sum of the quadratic residues modulo p

#quadratic residues #sum of squares #divisibility

Ex.1340Infinitely many triangular numbers that are squares

#Pell equation #recurrence #infinitely many

Diophantine equations

Ex.1341det(I + AB) = det(I + BA) by density

#density #determinant #polynomial identity

Applications and the case n = 2

Ex.1342Exponential coordinates on SO(3)

#exponential map #SO(3) #Rodrigues formula #homeomorphism

Applications and the case n = 2

Ex.1343Rank-one orthogonal projections form the projective space

#projective space #projections #compact Hausdorff #homeomorphism

Applications and the case n = 2

Ex.1344Odd-dimensional rotations have an axis

#orthogonal group #determinant #eigenvalue 1 #Euler's theorem

Applications and the case n = 2

Ex.1345The quotient of M_n(R) by conjugation is not Hausdorff

#quotient topology #conjugation #Jordan block #separation axioms

Applications and the case n = 2

Ex.1346AAáµ€ and Aáµ€A have the same characteristic polynomial

#characteristic polynomial #density #symmetric matrices

Applications and the case n = 2

Ex.1347The Grassmannian of orthogonal projections is compact and connected

#Grassmannian #compactness #connectedness #SO(n) orbit

Applications and the case n = 2

Ex.1348Sym_n/O(n) is a closed Weyl chamber

#orbit space #eigenvalues #Weyl chamber #quotient

Applications and the case n = 2

Ex.1349The rotation angle of a 3D rotation is continuous

#SO(3) #rotation angle #trace #normal form

Applications and the case n = 2

Ex.1350AB and BA have the same characteristic polynomial

#characteristic polynomial #density #similarity

Density and closures

Ex.1351Adjugate identities by density

#adjugate #determinant #density

Density and closures

Ex.1352Proper algebraic subsets of M_n(R) have measure zero

#Lebesgue measure #polynomials #Fubini #induction

Density and closures

Ex.1353The Cayley-Hamilton theorem by density

#Cayley-Hamilton #density #diagonalisation

Density and closures

Ex.1354Closure of the matrices with a non-real eigenvalue

#discriminant #non-real eigenvalues #closure #odd dimension

Density and closures

Ex.1355Finite-order rotations are dense in SO(2)

#rotations #roots of unity #density

Density and closures

Ex.1356Idempotents of a given rank form a closed set

#idempotent #projections #trace #rank

Density and closures

Ex.1357Powers of a rotation: when are they dense?

#Kronecker #pigeonhole #rotation group #dense subgroup

Density and closures

Ex.1358Matrices with an eigenvalue of modulus one

#spectrum #closed sets #empty interior #scaling

Density and closures

Ex.1359Matrices with an LU factorisation

#LU decomposition #leading principal minors #density

Density and closures

Ex.1360Nonderogatory matrices form an open dense set

#minimal polynomial #nonderogatory #density

Density and closures

Ex.1361Orthogonally invariant functions of symmetric matrices

#spectral theorem #invariant functions #eigenvalues

Density and closures

Ex.1362density in the special linear group

#special linear group #transvections #density

Density and closures

Ex.1363Rational rotation matrices are dense in SO(2)

#rational points #circle #stereographic projection #Pythagorean triples

Density and closures

Ex.1364The closure of the Hurwitz matrices

#Hurwitz #stability #spectrum #closure

Density and closures

Ex.1365The complement of a polynomial zero set is open and dense

#polynomials #density #Taylor expansion

Density and closures

Ex.1366The orbit of a Jordan block accumulates on the identity

#conjugacy orbit #closure #Jordan block

Density and closures

Ex.1367The set of real normal matrices

#normal matrices #closed sets #subspaces

Density and closures

Ex.1368A matrix of positive determinant that is not an exponential

#matrix exponential #surjectivity #commutation #diagonal matrices

Exponential, logarithm and group topology

Ex.1369Centres of GL_n and SL_n, and discrete normal subgroups

#centre #discrete subgroup #connected group

Exponential, logarithm and group topology

Ex.1370The determinant of an exponential: det e^A = e^(tr A)

#matrix exponential #determinant #trace #triangularisation

Exponential, logarithm and group topology

Ex.1371exp is a homeomorphism from symmetric to positive definite matrices

#symmetric matrices #positive definite #homeomorphism #proper map

Exponential, logarithm and group topology

Ex.1372The quotients GL_n/O(n) and SL_n/SO(n)

#homogeneous space #quotient topology #positive definite #traceless

Exponential, logarithm and group topology

Ex.1373The polar-exponential decomposition of GL_n

#polar decomposition #homeomorphism #orthogonal group #GL_n

Exponential, logarithm and group topology

Ex.1374SO(n)/SO(n-1) is the sphere

#orbit map #quotient #sphere #compact Hausdorff

Exponential, logarithm and group topology

Ex.1375The exponential maps skew-symmetric matrices onto SO(n)

#orthogonal group #skew-symmetric #normal form #connectedness

Exponential, logarithm and group topology

Ex.1376The logarithm near the identity and local inverse of exp

#matrix logarithm #local homeomorphism #power series

Exponential, logarithm and group topology

Ex.1377Connected components of the projective linear group

#projective group #quotient #determinant #parity

Exponential, logarithm and group topology

Ex.1378Nilpotent matrices and unipotent matrices

#nilpotent #unipotent #polynomial inverse #exponential

Exponential, logarithm and group topology

Ex.1379Topology of the groups of upper triangular matrices

#triangular matrices #connected components #contractible #unipotent

Exponential, logarithm and group topology

Ex.1380Eigenvalues are only Hölder continuous

#eigenvalues #Hölder continuity #Jordan block #perturbation

Rank, spectrum and norms

Ex.1381Semicontinuity of the inertia of symmetric matrices

#inertia #symmetric matrices #Weyl #semicontinuity

Rank, spectrum and norms

Ex.1382Inversion reverses the order of positive definite matrices

#Loewner order #positive definite #inverse #square root

Rank, spectrum and norms

Ex.1383Matrices with positive definite symmetric part

#positive definite #convex cone #eigenvalues #real part

Rank, spectrum and norms

Ex.1384Power-bounded matrices do not form a closed set

#power bounded #Jordan block #closed sets

Rank, spectrum and norms

Ex.1385Power sums determine the characteristic polynomial

#Newton identities #power sums #characteristic polynomial #continuity

Rank, spectrum and norms

Ex.1386The cone of rank-one matrices

#rank one #closed cone #connectedness #minors

Rank, spectrum and norms

Ex.1387Semicontinuity of the minimal polynomial degree and of the centraliser

#lower semicontinuity #minimal polynomial #centraliser #rank

Rank, spectrum and norms

Ex.1388Where is {det = 0} a manifold?

#determinant #implicit function theorem #singular point #cone #local homology

Rank, spectrum and norms

Ex.1389The discriminant of a 2×2 matrix and a cone

#discriminant #2x2 matrices #convex cone #complex eigenvalues

Rank, spectrum and norms

Ex.1390The set of matrices whose powers converge is neither open nor closed

#powers #convergence #open sets #closed sets

Rank, spectrum and norms

Ex.1391The spectral mapping theorem for polynomials

#spectrum #triangularisation #polynomial of a matrix

Rank, spectrum and norms

Ex.1392A^k converges to 0 if and only if the spectral radius is less than 1

#spectral radius #Jordan form #powers of a matrix

Rank, spectrum and norms

Ex.1393Three logarithmic series

#logarithmic series #telescoping #conditional convergence

Ex.1394Six numeric series: convergence study

#ratio test #root test #comparison #absolute convergence

Convergence tests

Ex.1395Three convergent series with explicit sums

#geometric series #telescoping #arctangent identity #d'Alembert

Ex.1396Three series with computed sums

#geometric series #telescoping #arctan identity

Ex.1397The operator norm via the largest eigenvalue of Aáµ€A

#spectral norm #singular values #Rayleigh quotient

Norms and metric structure

Ex.1398The operator norm of a shear matrix

#operator norm #golden ratio #singular values

Norms and metric structure

Ex.1399The operator norm is invariant under transposition

#transpose #operator norm #duality

Norms and metric structure

Ex.1400Eigenvalues are bounded by the operator norm

#spectral radius #complex eigenvalue #operator norm

Norms and metric structure

Ex.1401The Neumann series

#Neumann series #invertibility #geometric series #Banach algebra

Norms and metric structure

Ex.1402Invertibility is stable under small perturbations

#invertible matrices #perturbation #Neumann series #open set

Norms and metric structure

Ex.1403The trace is bounded by the Frobenius norm

#trace #Cauchy-Schwarz #Frobenius norm

Norms and metric structure

Ex.1404The Schur test

#Schur test #spectral radius #induced norms

Norms and metric structure

Ex.1405Strict convexity of the Frobenius ball but not of the operator ball

#strict convexity #unit ball #parallelogram law #operator norm

Norms and metric structure

Ex.1406The rank metric

#rank #discrete topology #metric #continuity

Norms and metric structure

Ex.1407Linear functionals on M_n(R) are traces against a matrix

#linear functionals #Riesz representation #continuity

Norms and metric structure

Ex.1408Induced operator norms make M_n(R) a Banach algebra

#Banach algebra #induced norm #equivalent norms

Norms and metric structure

Ex.1409All norms on M_n(R) are equivalent

#equivalence of norms #compactness #finite dimension

Norms and metric structure

Ex.1410Distance from an invertible matrix to the singular matrices

#distance to singular matrices #singular values #Eckarte-Young #SVD

Norms and metric structure

Ex.1411The inverse blows up near singular matrices

#inverse #blow-up #distance to singular matrices

Norms and metric structure

Ex.1412The best constants comparing operator and Frobenius norms

#equivalence constants #singular values #rank one #orthogonal

Norms and metric structure

Ex.1413The closure of {det > 0}

#determinant #singular value decomposition #closure #boundary

Open, closed, interior, boundary

Ex.1414Lower-rank matrices form a closed set, higher-rank an open one

#rank #minors #semicontinuity #nowhere dense

Open, closed, interior, boundary

Ex.1415Matrices of rank exactly r: neither open nor closed

#rank #closure #SVD #semicontinuity

Open, closed, interior, boundary

Ex.1416Matrices with a real eigenvalue: closed set, interior for n = 2

#real eigenvalues #closed sets #discriminant #2x2 matrices

Open, closed, interior, boundary

Ex.1417Real-diagonalisable matrices: neither open nor closed

#diagonalisable #Jordan blocks #counterexamples

Open, closed, interior, boundary

Ex.1418Matrices with n distinct eigenvalues form an open set

#discriminant #distinct eigenvalues #open sets #characteristic polynomial

Open, closed, interior, boundary

Ex.1419The orthogonal group is closed and nowhere dense

#orthogonal group #spheres #nowhere dense #Frobenius norm

Open, closed, interior, boundary

Ex.1420Involutions form a closed unbounded set

#involutions #closed sets #unbounded #counterexample

Open, closed, interior, boundary

Ex.1421The boundary of the positive definite cone

#positive definite #boundary #symmetric matrices #determinant

Open, closed, interior, boundary

Ex.1422Orthogonal projections: compact with empty interior in Sym_n

#projections #compactness #symmetric matrices #nowhere dense

Open, closed, interior, boundary

Ex.1423An open subgroup of GL_n is also closed

#topological groups #open subgroup #cosets #GL_n

Open, closed, interior, boundary

Ex.1424Interior and boundary of {det > 1}

#determinant #interior #boundary #level sets

Open, closed, interior, boundary

Ex.1425The commutant of a non-scalar matrix is a thin subspace

#commutant #subspaces #scalar matrices #nowhere dense

Open, closed, interior, boundary

Ex.1426Matrices with rational trace: dense with empty interior

#trace #density #rationals #open maps

Open, closed, interior, boundary

Ex.1427Countably many proper subspaces never cover M_n

#Baire category theorem #subspaces #covering

Open, closed, interior, boundary

Ex.1428The derivative of the determinant (Jacobi's formula)

#derivative of determinant #Jacobi formula #adjugate #density

Continuity of matrix maps

Ex.1429The derivative of matrix inversion

#derivative #inversion #Neumann series #Fréchet derivative

Continuity of matrix maps

Ex.1430Roots of a polynomial depend continuously on its coefficients

#roots of polynomials #Cauchy bound #Hölder continuity #characteristic polynomial

Continuity of matrix maps

Ex.1431The spectral radius is continuous

#spectral radius #continuity of eigenvalues #Hölder #characteristic polynomial

Continuity of matrix maps

Ex.1432Rank is lower but not upper semicontinuous

#rank #semicontinuity #minors #open sets

Continuity of matrix maps

Ex.1433The rank of the adjugate matrix

#adjugate #rank #minors #kernel

Continuity of matrix maps

Ex.1434The sign of the determinant under squaring

#squaring #determinant sign #open sets #GL_n^+

Continuity of matrix maps

Ex.1435The determinant is an open map

#determinant #open map #SVD #singular matrices

Continuity of matrix maps

Ex.1436The complement of the special linear group

#determinant #level sets #components #SL_n

Ex.1437Matrices of fixed rank

#rank #singular value decomposition #path connected

Ex.1438SO(n) is connected by Givens rotations

#Givens rotations #induction #special orthogonal group

Ex.1439Components of the sets of idempotents and involutions

#idempotent #involution #trace #rank #conjugacy orbits

Ex.1440Matrices of rank at least k are path connected

#rank #singular values #determinant sign change #path connected

Ex.1441The special linear group is path connected (transvections)

#special linear group #elementary matrices #transvections #row reduction

Ex.1442The special linear group is path connected (polar decomposition)

#special linear group #polar decomposition #path connected

Ex.1443SO(n) is path connected (normal form)

#special orthogonal group #normal form #rotations #path connected

Ex.1444Continuity of a piecewise function with an essential singularity

#piecewise #continuity #exponential singularity

Ex.1445Complex coordinates (z, w) on M_2(R): determinant and singular values

#M_2(R) #complex coordinates #singular values #determinant #Frobenius norm

Applications and the case n = 2

Ex.1446Conjugation by rotations acts on w by e^{2iÎ²}

#M_2 #conjugation #rotations #complex coordinates #invariants

Applications and the case n = 2

Ex.1447Matrices of finite order are dense in SO(n)

#SO(n) #finite order #normal form of orthogonal matrices #density

Applications and the case n = 2

Ex.1448Real eigenvalues of a matrix

#M_2 #eigenvalues #discriminant #complex coordinates #real spectrum

Applications and the case n = 2

Ex.1449Rotations of finite order are dense in SO(3)

#rotations #SO(3) #density #Euler's theorem #finite order

Applications and the case n = 2

Ex.1450Signature is constant on a rank stratum of symmetric matrices

#signature #Sylvester's law of inertia #connected components #symmetric matrices

Applications and the case n = 2

Ex.1451The distance to the singular matrices in M_2(R)

#M_2 #distance to singular matrices #smallest singular value #operator norm

Applications and the case n = 2

Ex.1452The singular matrices of M_2(R) form a cone over a torus

#M_2 #singular matrices #rank one #torus #cone

Applications and the case n = 2

Ex.1453The shapes of GL(2,R), SL(2,R) and {det < 0}

#M_2 #polar coordinates #homeomorphism #circle #GL_2 #SL_2

Applications and the case n = 2

Ex.1454The operator-norm ball and O(2) in complex coordinates

#M_2 #operator norm #extreme points #O(2) #unit ball

Applications and the case n = 2

Ex.1455Schur-stable 2×2 matrices: the Jury conditions and connectedness

#spectral radius #Schur stability #Jury criterion #M2(R) #path connectedness

Applications and the case n = 2

Ex.1456The cone of positive definite 2×2 matrices is a light cone

#positive definite matrices #convex cone #Sym2 #boundary

Applications and the case n = 2

Ex.1457The convex hull of SO(2) is a flat disc

#M2(R) #convex hull #rotations #complex coordinates

Applications and the case n = 2

Ex.1458The 2×2 Hurwitz matrices form a path-connected set

#M2(R) #Hurwitz stability #path connectedness #trace and determinant

Applications and the case n = 2

Ex.1459Rank-one idempotents and involutions of M_2(R) are cylinders

#M_2(R) #complex coordinates #idempotents #involutions #trace and determinant

Applications and the case n = 2

Ex.1460The distance from a 2×2 matrix to the rotation group

#nearest rotation #Frobenius norm #SO(2) #M2(R) #Procrustes

Applications and the case n = 2

Ex.1461The nonzero nilpotents of M_2(R) form two components

#M_2(R) #nilpotent matrices #connected components #complex coordinates

Applications and the case n = 2

Ex.1462The square roots of A in M_2(R) form two disjoint planes

#M_2(R) #complex coordinates #complex structures #connected components

Applications and the case n = 2

Ex.1463Cantor's theorem

#diagonal argument #sets

Countability and cardinality

Ex.1464How fast does (1 + x)^(1/x) reach e?

#number e #Taylor expansion #order of convergence

Ex.1465Where power means lead

#power mean #geometric mean #logarithm #expansions

Ex.1466Reciprocal squares of x and tangent

#Taylor expansion #infinity minus infinity #tangent

Ex.1467The Dirichlet function has no limit anywhere

#sequences #density #non-existence of limits #pathological functions

Ex.1468A recurrence with a double fixed point

#fixed points #monotone sequences #parameter dependence

Ex.1469A functional equation solved with derivatives

#functional equations #differentiation #polynomials

Differentiable functions

Ex.1470The series of 1/(ln n)^(ln n)

#comparison test #exponential form #logarithm

Convergence tests

Ex.1471Adjacent sequences and the irrationality of e

#adjacent sequences #number e #irrationality

Ex.1472Bounds for n! by comparing with an integral

#factorial #integral comparison #Stirling

Riemann integral

Ex.1473Cauchy's condensation test

#condensation test #p-series #monotone terms

Convergence tests

Ex.1474A Cauchy product that diverges

#Cauchy product #conditional convergence #Mertens theorem

Convergence tests

Ex.1475Cauchy-Schwarz inequality via optimisation

#Cauchy-Schwarz #optimisation #proof

Optimisation in several variables

Ex.1476Constrained extrema and a compactness argument

#Lagrange #compactness #global maximum

Optimisation in several variables

Ex.1477Maximizing a product under a linear constraint

#Lagrange #product #linear constraint

Optimisation in several variables

Ex.1478Least squares as an optimisation problem

#least squares #optimisation #regression

Optimisation in several variables

Ex.1479Continuous injective functions are monotone

#intermediate value theorem #injectivity #monotonicity

Intermediate Value Theorem

Ex.1480A convex function and its asymptote

#convexity #slopes #asymptotes

Differentiable functions

Ex.1481Convexity forces the derivative to be continuous

#convexity #Darboux theorem #monotone functions #continuity

Differentiable functions

Ex.1482Existence of a Bézout identity

#Bézout #existence #well-ordering

Divisibility and gcd

Ex.1483A Chinese remainder system

#Chinese remainder theorem #congruences #Bézout

Ex.1484A common divisor divides 43

#divisibility #linear combination #prime number

Divisibility and gcd

Ex.1485Solving a linear Diophantine equation

#Diophantine equation #Bézout #particular solution

Diophantine equations

Ex.1486Solving $ax+by=c$

#Diophantine equation #Bézout #solvability criterion

Diophantine equations

Ex.1487Solving $\divisibility(a,b)+\operatorname{lcm}(a,b)=9+b$

#divisibility #lcm #Diophantine equation

Divisibility and gcd

Ex.1488A prime divisor property

#primes #divisibility #Euclid's lemma

Divisibility and gcd

Ex.1489An integrating factor

#integrating factor #linear equation #exact equation

First-order equations

Ex.1490The generating function of Fibonacci numbers

#generating function #Fibonacci #partial fractions #radius of convergence

Ex.1491How many solutions does ln x = ax have?

#intermediate value theorem #parameters #logarithm

Intermediate Value Theorem

Ex.1492Integrability with one wild point

#Riemann integrability #Darboux criterion #oscillation

Riemann integral

Ex.1493Limits of integrals depending on n

#limits of integrals #squeeze theorem #integration by parts

Riemann integral

Ex.1494Complex eigenvalues

#complex eigenvalues #real solutions #coefficients ind�termin�s

Ex.1495Matrix exponential with a nilpotent part

#matrix exponential #nilpotent #Jordan form

Ex.1496A linear system with a second member

#second member #variation of constants #matrix exponential

Ex.1497Affine change of variables

#change of variables #Jacobian #parallelogram

Multiple integrals

Ex.1498Is a function $C^1$ on $\mathbb{R}^2$?

#continuity #partial derivatives #class C1

Partial derivatives

Ex.1499A series of functions

#series of functions #normal convergence #pointwise convergence

Ex.1500Resonance with a double root

#resonance #double root #particular solution #identification

Linear equations of order two

Ex.1501A second-order equation with a polynomial right-hand side

#particular solution #identification #polynomial right-hand side

Linear equations of order two

Ex.1502The sequence sin(n) does not converge

#divergence #trigonometry #subsequences #contradiction

Ex.1503Speed of convergence

#speed of convergence #fixed point #mean value theorem

Ex.1504The alternating harmonic sums converge

#adjacent subsequences #alternating series #ln 2

Ex.1505The arithmetic–geometric mean

#adjacent sequences #AM-GM #means

Ex.1506The generating function of central binomial coefficients

#binomial series #generalised binomial coefficients #radius of convergence

Ex.1507The Cauchy criterion

#Cauchy sequences #completeness #series

Ex.1508A ratio test that fails, and how to conclude

#ratio test #central binomial coefficients #induction #Stirling

Convergence tests

Ex.1509The Cesàro lemma

#Cesaro mean #epsilon-N #averages

Ex.1510The Euler-Mascheroni constant as a series

#Euler-Mascheroni constant #harmonic numbers #telescoping #logarithm

Ex.1511e^x e^y = e^(x+y) from power series

#Cauchy product #exponential series #binomial theorem

Ex.1512The horizontal chord of length one half

#intermediate value theorem #chords #clever auxiliary function

Intermediate Value Theorem

Ex.1513The longest ladder around a corner

#optimisation #trigonometry #geometry #applications

Optimisation and curve sketching

Ex.1514From the ratio to the n-th root

#Cesaro lemma #n-th root #ratio #Stirling

Ex.1515The ratio test with a parameter

#ratio test #Stirling formula #critical case

Convergence tests

Ex.1516Monotonicity of (1 + 1/n)^n

#number e #AM-GM inequality #monotone convergence

Ex.1517The tangent half-angle substitution

#Weierstrass substitution #trigonometric integrals #rational functions

Integration techniques

Ex.1518Divergence theorem (Ostrogradsky)

#divergence theorem #flux #surface

Vector calculus

Ex.1519Green-Riemann formula

#Green-Riemann #line integral #double integral

Vector calculus

Ex.1520Stokes' theorem

#Stokes #curl #surface integral

Vector calculus

Ex.1521Wallis integrals

#Wallis integrals #recurrence #asymptotics #pi

Integration techniques

Ex.1522Differentiable but not continuously differentiable

#differentiability #continuity of the derivative #counterexamples

Differentiable functions

Ex.1523An expansion of order 2 without a second derivative

#Taylor-Young converse #counterexample #differentiability

Taylor expansions

Ex.1524A rational approximation of cosine

#Padé approximant #undetermined coefficients #approximation

Taylor expansions

Ex.1525Expanding the inverse of x + x³

#Taylor expansion #inverse function #fixed point iteration

Taylor expansions

Ex.1526Binomial coefficient inverses

#inverse #binomial #sum

Binomial coefficients

Ex.1527An identity with inverses

#inverse binomial #identity #integral

Binomial coefficients

Ex.1528Catalan via binomial

#Catalan #binomial #generating function

Binomial coefficients

Ex.1529Central binomial coefficients

#central binomial #asymptotics #Wallis

Binomial coefficients

Ex.1530Choosing subsets of subsets

#subsets #binomial #identity

Binomial coefficients

Ex.1531Binomial coefficients and primes

#prime #binomial #divisibility

Binomial coefficients

Ex.1532A combinatorial sum

#combinatorial sum #double counting #identity

Binomial coefficients

Ex.1533A combinatorial sum with choice

#combinatorial sum #binomial #choice

Binomial coefficients

Ex.1534Cyclic identity with binomials

#cyclic #binomial #identity

Binomial coefficients

Ex.1535Approximation via binomial theorem

#approximation #binomial #sqrt

Binomial coefficients

Ex.1536Fibonacci and binomials

#Fibonacci #binomial #sum

Binomial coefficients

Ex.1537Generating function of binomial row

#generating function #binomial row #polynomial

Binomial coefficients

Ex.1538A complex binomial identity

#complex numbers #binomial #identity

Binomial coefficients

Ex.1539Sum of sines

#complex numbers #binomial #sine

Binomial coefficients

Ex.1540An identity with squares

#squares #binomial #double counting

Binomial coefficients

Ex.1541Application to integration

#integration #binomial #identity

Binomial coefficients

Ex.1542An integral identity

#integral #binomial #identity

Binomial coefficients

Ex.1543The multinomial theorem

#multinomial #expansion #coefficients

Binomial coefficients

Ex.1544Multinomial coefficients

#multinomial #coefficient #expansion

Binomial coefficients

Ex.1545Partial sums of binomials

#partial sum #binomial #hockey stick

Binomial coefficients

Ex.1546Hexagon identity

#hexagon #Pascal #binomial

Binomial coefficients

Ex.1547Pentagon identity

#pentagon #Pascal #binomial

Binomial coefficients

Ex.1548Weighted power sum

#weighted sum #binomial #derivative

Binomial coefficients

Ex.1549Quadratic coefficient sum

#quadratic #coefficient #binomial

Binomial coefficients

Ex.1550Moments of the binomial row

#moments #binomial row #mean

Binomial coefficients

Ex.1551Sum of squares of binomial coefficients

#square sum #Vandermonde #binomial

Binomial coefficients

Ex.1552Stirling numbers and binomials

#Stirling #binomial #identity

Binomial coefficients

Ex.1553Convolution of binomial rows

#convolution #binomial #Vandermonde

Binomial coefficients

Ex.1554Reciprocals of binomial coefficients

#reciprocal #sum #telescoping

Binomial coefficients

Ex.1555Sum with $k^2\binom{n}{k}$

#weighted sum #square #binomial

Binomial coefficients

Ex.1556A generalized Vandermonde

#Vandermonde #generalization #identity

Binomial coefficients

Ex.1557Bounding f′ by f and f″ (Landau's inequality)

#Taylor-Lagrange #inequalities #optimisation of a parameter

Taylor expansions

Ex.1558Cantor–Bernstein theorem

#Cantor-Bernstein #bijection #injections

Countability and cardinality

Ex.1559The Cantor set is uncountable

#Cantor set #uncountable #ternary

Countability and cardinality

Ex.1560Cantor's theorem

#Cantor theorem #power set #larger cardinality

Countability and cardinality

Ex.1561Finite subsets of $\mathbb{N}$

#finite subsets #countable #naturals

Countability and cardinality

Ex.1562Ordinals vs cardinals

#ordinals #cardinals #distinction

Countability and cardinality

Ex.1563$|\mathcal{P}(\mathbb{N})|=|\mathbb{R}|$

#power set #reals #cardinality

Countability and cardinality

Ex.1564$\mathbb{R}$ is uncountable

#reals #uncountable #Cantor diagonal

Countability and cardinality

Ex.1565Countable union of countable sets

#countable union #diagonalization #countable

Countability and cardinality

Ex.1566Central binomial coefficients from (1 − 4x)^(−1/2)

#binomial series #generalised binomial coefficients #central binomial

Taylor expansions

Ex.1567An inequality involving arguments

#argument #inequality #complex

Complex numbers

Ex.1568An ellipse via a modulus equation

#ellipse #locus #modulus

Complex numbers

Ex.1569Equilateral triangle condition

#equilateral triangle #complex #roots of unity

Complex numbers

Ex.1570A locus involving an angle

#locus #angle #complex

Complex numbers

Ex.1571A region in the complex plane

#region #inequality #geometric

Complex numbers

Ex.1572The modulus of a sum

#modulus #sum #inequality

Complex numbers

Ex.1573Product formula for cosines

#product formula #cosines #trigonometric

Complex numbers

Ex.1574Sum of a trigonometric series

#trigonometric series #complex #geometric

Complex numbers

Ex.1575Square roots of complex numbers

#square roots #complex #system

Complex numbers

Ex.1576Sum of cosines

#trigonometric sum #cosines #complex

Complex numbers

Ex.1577Trigonometric identities via complex numbers

#De Moivre #trigonometric identities #complex

Complex numbers

Ex.1578CRT with non-coprime moduli

#CRT #non-coprime #compatibility

Ex.1579The CRT as a ring isomorphism

#CRT #ring isomorphism #product

Ex.1580The CRT with three moduli

#CRT #three moduli #pairwise coprime

Ex.1581The Diffie–Hellman protocol

#Diffie-Hellman #cryptography #modular

Ex.1582Euler's theorem applied

#Euler theorem #totient #computation

Ex.1583Fermat's little theorem and pseudoprimes

#Fermat #pseudoprime #compositeness

Ex.1584Proving Fermat's little theorem

#Fermat #proof #permutation

Ex.1585Legendre symbol and Euler's criterion

#Legendre symbol #Euler criterion #quadratic reciprocity

Ex.1586RSA and modular arithmetic

#RSA #cryptography #modular exponentiation

Ex.1587Primitive roots

#primitive root #generator #cyclic group

Ex.1588Quadratic residues mod $p$

#quadratic residues #Legendre symbol #Euler criterion

Ex.1589A system of linear congruences

#system #linear congruences #CRT

Ex.1590Wilson's theorem

#Wilson theorem #prime #factorial

Ex.1591Cauchy–Schwarz via optimization

#Cauchy-Schwarz #optimization #inequality

Optimisation in several variables

Ex.1592Optimization under inequality constraints

#inequality constraint #KKT #boundary

Optimisation in several variables

Ex.1593Classifying a 3D critical point

#Hessian #3D #saddle

Optimisation in several variables

Ex.1594Three-variable saddle point

#three variables #saddle #Hessian

Optimisation in several variables

Ex.1595Shortest distance from a point to a line

#Lagrange #distance #line

Optimisation in several variables

Ex.1596Distance from origin to an ellipsoid

#Lagrange #ellipsoid #distance

Optimisation in several variables

Ex.1597Lagrange with exponential objective

#Lagrange #exponential #constrained maximum

Optimisation in several variables

Ex.1598Global extrema on a compact set

#compactness #global extrema #boundary

Optimisation in several variables

Ex.1599Maximizing a volume with fixed perimeter

#Lagrange #geometric optimization #volume

Optimisation in several variables

Ex.1600Constrained quadratic form

#quadratic form #Rayleigh quotient #eigenvalues

Optimisation in several variables

Ex.1601Image of a compact set under a continuous function

#compactness #continuous image #extrema

Ex.1602A function that is not continuous despite radial limits

#continuity #radial limit #counterexample

Ex.1603Generating function for binomial coefficients

#generating function #binomial #formal power series

Counting principles

Ex.1604Catalan applications

#Catalan #applications #structures

Counting principles

Ex.1605A combinatorial identity

#combinatorial identity #double counting #binomial

Counting principles

Ex.1606Derangements

#derangements #inclusion-exclusion #permutations

Counting principles

Ex.1607Double counting

#double counting #combinatorial proof #count in two ways

Counting principles

Ex.1608Another double count

#double counting #combinatorial identity #binomial

Counting principles

Ex.1609Alternating sum identity

#alternating sum #binomial #identity

Counting principles

Ex.1610Counting coprime integers

#inclusion-exclusion #coprime #totient

Counting principles

Ex.1611Derangements via inclusion-exclusion

#derangements #inclusion-exclusion #inclusion

Counting principles

Ex.1612Derangement formula

#derangement #inclusion-exclusion #formula

Counting principles

Ex.1613A combinatorial identity via inclusion-exclusion

#inclusion-exclusion #binomial identity

Counting principles

Ex.1614Surjection formula proof

#surjection #inclusion-exclusion #onto

Counting principles

Ex.1615Sieve formula

#sieve #inclusion-exclusion #primes

Counting principles

Ex.1616Surjections via inclusion-exclusion

#surjections #inclusion-exclusion #function counting

Counting principles

Ex.1617Surjections onto $3$ elements

#surjections #inclusion-exclusion #Stirling

Counting principles

Ex.1618Three-set inclusion-exclusion

#inclusion-exclusion #three sets #union

Counting principles

Ex.1619Three-set counts

#inclusion-exclusion #three sets #union

Counting principles

Ex.1620Counting with matrices

#matrices #counting #trace

Counting principles

Ex.1621The multinomial theorem

#multinomial #product rule #word counting

Counting principles

Ex.1622Lattice paths

#lattice paths #binomial coefficient #monotone paths

Counting principles

Ex.1623Permutations with fixed points

#fixed points #permutations #inclusion-exclusion

Counting principles

Ex.1624Pigeonhole with 100 numbers

#pigeonhole #subsets #olympiad

Counting principles

Ex.1625Multiples via pigeonhole

#pigeonhole #multiples #divisibility

Counting principles

Ex.1626Pigeonhole in a graph

#pigeonhole #graph #vertex degree

Counting principles

Ex.1627Pigeonhole with points

#pigeonhole #geometry #Dirichlet

Counting principles

Ex.1628Sum via pigeonhole

#pigeonhole #sums #olympiad

Counting principles

Ex.1629Counting surjections

#surjections #inclusion-exclusion #Stirling

Counting principles

Ex.1630Vandermonde's identity

#Vandermonde #double counting #binomial

Counting principles

Ex.1631Computing the inverse with cofactors

#cofactor #comatrix #inverse

Ex.1632The cofactor formula for the inverse

#cofactor #comatrix #inverse formula

Ex.1633Determinant of a companion matrix

#companion matrix #determinant #characteristic polynomial

Ex.1634Cramer's rule for a 3×3 system

#Cramer's rule #3x3 determinant #linear system

Ex.1635A cyclic 3×3 determinant

#cyclic determinant #factorisation #roots of unity

Ex.1636A mixed determinant exercise

#determinant #row operations #factorisation

Ex.1637The determinant as a volume

#determinant #volume #parallelepiped

Ex.1638A numerical 4×4 determinant

#4x4 determinant #row reduction #computation

Ex.1639Invertibility depending on a parameter

#parameter #invertibility #determinant

Ex.1640Determinant of a product

#determinant of product #multiplicativity #proof

Ex.1641Determinant of a skew-symmetric matrix

#skew-symmetric #determinant #odd dimension

Ex.1642Trace and determinant via eigenvalues

#characteristic polynomial #trace #determinant #eigenvalues

Ex.1643The Vandermonde determinant

#Vandermonde #determinant #factorisation

Ex.1644Egyptian fractions

#Egyptian fractions #unit fractions #greedy algorithm

Divisibility and gcd

Ex.1645Chinese remainder theorem

#Chinese remainder #congruences #CRT

Divisibility and gcd

Ex.1646Coprime numbers and exponents

#coprime #exponents #gcd

Divisibility and gcd

Ex.1647Coprime sums and products

#coprime #sum #product

Divisibility and gcd

Ex.1648An application of the Euclidean algorithm

#Euclidean algorithm #expansion #continued fraction

Divisibility and gcd

Ex.1649A Diophantine equation with coefficients

#Diophantine #Bézout #general solution

Divisibility and gcd

Ex.1650Euler's totient function

#Euler totient #coprime #formula

Divisibility and gcd

Ex.1651GCD of sums

#gcd #sums #congruence

Divisibility and gcd

Ex.1652GCD via the Euclidean matrix

#gcd #Euclidean algorithm #matrix

Divisibility and gcd

Ex.1653Divisors of Mersenne numbers

#Mersenne #divisors #congruences

Divisibility and gcd

Ex.1654GCD in $\mathbb{Z}$ vs $\mathbb{R}[X]$

#analogy #gcd #polynomials #integers

Divisibility and gcd

Ex.1655GCD of polynomial coefficients

#gcd #coefficients #polynomials

Divisibility and gcd

Ex.1656Primitive Pythagorean triples

#Pythagorean triples #coprime #parity

Divisibility and gcd

Ex.1657e is irrational

#Taylor-Lagrange #irrationality #exponential

Taylor expansions

Ex.1658Functions of a diagonalizable matrix

#functions of matrices #diagonalization #polynomial

Eigenvalues and diagonalisation

Ex.1659The diagonalizability criterion

#diagonalizability #multiplicity #eigenspaces

Eigenvalues and diagonalisation

Ex.1660An involution is diagonalizable

#involution #diagonalization #eigenvalues

Eigenvalues and diagonalisation

Ex.1661A projection is diagonalizable

#projection #diagonalization #eigenvalues

Eigenvalues and diagonalisation

Ex.1662Diagonalizing a 2×2 matrix

#diagonalization #change of basis #eigenvalues

Eigenvalues and diagonalisation

Ex.1663Diagonalizing a 3×3 matrix

#diagonalization #eigenspaces #change of basis

Eigenvalues and diagonalisation

Ex.1664Geometric versus algebraic multiplicity

#multiplicities #eigenspace #diagonalization

Eigenvalues and diagonalisation

Ex.1665A matrix with a single eigenvalue

#eigenvalues #eigenspace #diagonalization

Eigenvalues and diagonalisation

Ex.1666A cubic equation with parameter

#cubic #parameter #roots

Equations and radicals

Ex.1667A cyclotomic equation

#roots of unity #cyclotomic #complex

Equations and radicals

Ex.1668A double radical

#double radical #squaring #verification

Equations and radicals

Ex.1669A functional equation

#functional equation #polynomial solutions #algebraic

Equations and radicals

Ex.1670An equation with the floor function

#floor function #integer part #olympiad

Equations and radicals

Ex.1671A radical equation with parameter

#parameter #radical equation #conditions

Equations and radicals

Ex.1672Polynomial divisibility and roots

#polynomial #divisibility #factor theorem

Equations and radicals

Ex.1673A polynomial equation with parameter

#parameter #roots #discriminant

Equations and radicals

Ex.1674Counting real roots

#polynomial #real roots #derivative

Equations and radicals

Ex.1675A quadratic with parameter

#parameter #quadratic #discriminant

Equations and radicals

Ex.1676A quartic with no real roots

#quartic #real roots #sum of squares

Equations and radicals

Ex.1677A radical equation leading to a quartic

#radical #quartic #substitution #extraneous roots

Equations and radicals

Ex.1678A system with an inequality

#system #inequality #integers

Equations and radicals

Ex.1679A system depending on a parameter

#parameter #linear system #discussion

Equations and radicals

Ex.1680A quadratic system

#nonlinear system #elimination #olympiad

Equations and radicals

Ex.1681Planar bipartite bound

#bipartite #planar #edge bound

Ex.1682Chromatic number

#chromatic number #coloring #clique

Ex.1683Graphical degree sequences

#degree sequence #graphical #Erdős–Gallai

Ex.1684Dijkstra's algorithm

#Dijkstra #shortest path #weighted graph

Ex.1685Eulerian circuits

#Eulerian #Euler circuit #degree

Ex.1686The four-color theorem (statement)

#four-color #planar #chromatic

Ex.1687Hall's marriage theorem (statement)

#Hall #marriage #matching

Ex.1688$K_{3,3}$ is not planar

#bipartite #planarity #K33

Ex.1689Kőnig's theorem

#Kőnig #bipartite #matching #vertex cover

Ex.1690Kruskal's algorithm

#Kruskal #MST #greedy

Ex.1691Minimum degree and connectivity

#minimum degree #connectivity #edge connectivity

Ex.1692Moore bound

#Moore bound #diameter #degree

Ex.1693NP-hard problems in graphs

#NP-hard #Hamiltonian #coloring

Ex.1694Planar graphs and Euler's formula

#planar #Euler formula #faces

Ex.1695Prim's algorithm

#Prim #MST #greedy

Ex.1696Prüfer sequences

#Prüfer #bijection #labeled trees

Ex.1697Ramsey numbers

#Ramsey #R(3 #3) #coloring

Ex.1698Cayley's formula

#Cayley #labeled trees #count

Ex.1699Cauchy's theorem applied to $A_4$

#Cauchy #alternating group #elements of order

Ex.1700Cauchy's theorem for groups

#Cauchy theorem #prime order #existence

Ex.1701The Chinese remainder theorem for groups

#Chinese remainder #direct product #cyclic group

Ex.1702Subgroups of a cyclic group

#cyclic group #subgroups #divisors

Ex.1703The dihedral group $D_n$

#dihedral group #symmetries #presentation

Ex.1704The first isomorphism theorem

#first isomorphism theorem #quotient #kernel

Ex.1705The order of $GL_n(\mathbb{F}_p)$

#general linear group #finite field #order

Ex.1706Normal subgroups

#normal subgroup #conjugation #criterion

Ex.1707The normalizer

#normalizer #subgroup #normal

Ex.1708Quotient groups

#quotient group #normal subgroup #cosets

Ex.1709The sign homomorphism

#sign #parity #homomorphism

Ex.1710Simple groups

#simple group #normal subgroup #examples

Ex.1711The special linear group $SL_n(\mathbb{R})$

#special linear group #determinant #normal subgroup

Ex.1712Four-set inclusion-exclusion

#IE #four sets #union

Inclusion-exclusion

Ex.1713Application to probability

#IE #probability #events

Inclusion-exclusion

Ex.1714At least two properties

#IE #at least two #intersection

Inclusion-exclusion

Ex.1715Inclusion-exclusion in ballot problems

#IE #ballot #paths

Inclusion-exclusion

Ex.1716Colorings avoiding patterns

#IE #coloring #grid

Inclusion-exclusion

Ex.1717Numbers coprime to $210$

#IE #coprime #totient

Inclusion-exclusion

Ex.1718Path counting via IE

#IE #paths #obstacles

Inclusion-exclusion

Ex.1719Derangements via double counting

#derangements #double counting #IE

Inclusion-exclusion

Ex.1720Pairwise but not triple

#IE #triple #void

Inclusion-exclusion

Ex.1721A multi-property application

#IE #application #multi

Inclusion-exclusion

Ex.1722Committees by inclusion-exclusion

#IE #committees #at least one

Inclusion-exclusion

Ex.1723Combining multiple IE applications

#IE #combining #multi-step

Inclusion-exclusion

Ex.1724Probability a permutation has a fixed point

#IE #probability #fixed point

Inclusion-exclusion

Ex.1725Proof of inclusion-exclusion

#inclusion-exclusion #proof #indicator

Inclusion-exclusion

Ex.1726Sieve formula

#IE #sieve #formula

Inclusion-exclusion

Ex.1727Colorings with constraints

#IE #coloring #constraints

Inclusion-exclusion

Ex.1728Avoiding two letters

#IE #words #avoiding letters

Inclusion-exclusion

Ex.1729Two obstacles via IE

#IE #paths #obstacles

Inclusion-exclusion

Ex.1730Words avoiding letters

#IE #words #letters

Inclusion-exclusion

Ex.1731The weighted AM-GM inequality

#weighted AM-GM #inequality #convexity

Ex.1732Cauchy-Schwarz and Jensen

#Cauchy-Schwarz #Jensen #convexity

Ex.1733Chebyshev's sum inequality

#Chebyshev #sum inequality #monotone sequences

Ex.1734Nesbitt's inequality

#Nesbitt #inequality #fractions

Ex.1735Nesbitt's inequality via Cauchy-Schwarz

#Nesbitt #Cauchy-Schwarz #Engel form

Ex.1736Power mean inequality

#power mean #inequalities #means

Ex.1737Positivity of a quadratic expression

#positivity #discriminant #inequalities

Ex.1738Bounding a sum of square roots

#square roots #AM-GM #inequalities

Ex.1739A ratio-sum inequality

#ratios #AM-GM #cyclic sums

Ex.1740An inequality by substitution

#substitution #AM-GM #inequalities

Ex.1741Sum of square roots

#square roots #Cauchy-Schwarz #inequalities

Ex.1742Cauchy-Schwarz via sums of squares

#Cauchy-Schwarz #sums of squares #algebraic proof

Ex.1743Cyclic sum of squares

#cyclic sum #squares #factorization

Ex.1744Minkowski's inequality

#Minkowski #triangle inequality #norms

Ex.1745Adjoint of a linear map

#adjoint #linear map #inner product

Inner product spaces

Ex.1746Bessel's inequality

#Bessel inequality #orthonormal family #projection

Inner product spaces

Ex.1747Distance to a subspace

#distance #orthogonal projection #minimization

Inner product spaces

Ex.1748Orthogonal projection onto a subspace

#orthogonal projection #subspace #orthonormal basis

Inner product spaces

Ex.1749Completing an orthonormal family

#orthonormal family #basis #completion

Inner product spaces

Ex.1750Parseval's identity

#Parseval identity #orthonormal basis #norm

Inner product spaces

Ex.1751Self-adjoint operators and symmetric matrices

#self-adjoint #symmetric matrix #eigenvalues

Inner product spaces

Ex.1752$e$ is not a quadratic irrational

#e #quadratic irrational #contradiction

Ex.1753Liouville's theorem and examples

#Liouville #approximation #transcendental

Ex.1754Independence of square roots

#independence #square roots #linear independence

Ex.1755Euler characteristic by induction

#induction #Euler characteristic #trees

Ex.1756Determinant of a matrix by induction

#induction #matrix #determinant

Ex.1757Existence of quotient and remainder

#strong induction #division #quotient

Ex.1758Every positive integer is a sum of non-consecutive Fibonacci numbers

#induction #Fibonacci #Zeckendorf

Ex.1759Representation of integers

#strong induction #representation #powers

Ex.1760Tiling by induction

#induction #tiling #L-tromino

Ex.1761Russell's paradox (statement)

#Russell paradox #sets #self-reference

Proof techniques

Ex.1762The Cayley-Hamilton theorem

#Cayley-Hamilton #characteristic polynomial #matrix powers

Matrices and linear systems

Ex.1763Cholesky factorization of a 2×2 matrix

#Cholesky #positive definite #factorization

Matrices and linear systems

Ex.1764Diagonalizing a 2×2 matrix

#diagonalization #eigenvalues #change of basis

Matrices and linear systems

Ex.1765Diagonalizing a 3×3 matrix

#diagonalization #eigenvalues #eigenvectors

Matrices and linear systems

Ex.1766When is a matrix diagonalizable?

#diagonalization #eigenspaces #multiplicity

Matrices and linear systems

Ex.1767Eigenvalues of a 3×3 matrix

#eigenvalues #characteristic polynomial #trace and determinant

Matrices and linear systems

Ex.1768Inverse of a 3×3 matrix by row reduction

#inverse #Gaussian elimination #3x3 matrices

Matrices and linear systems

Ex.1769A linear recurrence and matrix powers

#linear recurrence #diagonalization #matrix powers

Matrices and linear systems

Ex.1770LU decomposition

#LU decomposition #Gaussian elimination #factorization

Matrices and linear systems

Ex.1771The minimal polynomial of a matrix

#minimal polynomial #characteristic polynomial #diagonalizability

Matrices and linear systems

Ex.1772Orthogonal projection onto a subspace

#orthogonal projection #projection matrix #orthonormal basis

Matrices and linear systems

Ex.1773Positive definiteness of a symmetric matrix

#positive definite #Sylvester criterion #eigenvalues

Matrices and linear systems

Ex.1774Powers of a companion matrix

#companion matrix #powers #recurrence

Matrices and linear systems

Ex.1775Reducing a quadratic form

#quadratic form #reduction #completing the square

Matrices and linear systems

Ex.1776Signature of a quadratic form

#quadratic form #signature #diagonalization

Matrices and linear systems

Ex.1777Rank depending on a parameter

#rank #parameter #Gaussian elimination

Matrices and linear systems

Ex.1778A linear system with a parameter

#linear system #parameter #discussion

Matrices and linear systems

Ex.1779Baire category theorem

#Baire #category #complete

Metric and normed spaces

Ex.1780Completion of a metric space

#completion #Cauchy #dense embedding

Metric and normed spaces

Ex.1781Closed bounded sets need not be compact

#counterexample #compactness #infinite dimension

Metric and normed spaces

Ex.1782Heine–Cantor theorem

#Heine-Cantor #uniform continuity #compact

Metric and normed spaces

Ex.1783Centre of gravity of a planar region

#centroid #parabola #uniform density

Multiple integrals

Ex.1784Linear change of variables

#change of variables #Jacobian #triangle

Multiple integrals

Ex.1785Change of variables to simplify the integrand

#change of variables #Jacobian #triangular

Multiple integrals

Ex.17863D change of variables

#change of variables #3D #Jacobian

Multiple integrals

Ex.1787Volume using cylindrical coordinates

#cylindrical coordinates #volume #cone

Multiple integrals

Ex.1788The Fubini trick

#Fubini #double integral #iterated

Multiple integrals

Ex.1789Polar region between two circles

#polar #circle #region between circles

Multiple integrals

Ex.1790Spherical coordinates with a radial integrand

#spherical coordinates #radial #integration

Multiple integrals

Ex.1791Volume of a sphere by triple integral

#spherical coordinates #volume #sphere

Multiple integrals

Ex.1792Triple integral over a cube

#cube #triple integral #iterated

Multiple integrals

Ex.1793Triple integral over a simplex with cube

#simplex #triple integral #iterated

Multiple integrals

Ex.1794Triple integral with sine

#triple integral #sine #simplex

Multiple integrals

Ex.1795Differentiability of a piecewise function

#differentiability #continuity #piecewise

Partial derivatives

Ex.1796The Hessian matrix

#Hessian #second derivatives #quadratic form

Optimisation in several variables

Ex.1797Implicit differentiation

#implicit differentiation #Implicit Function Theorem

Partial derivatives

Ex.1798Inverse function theorem (illustration)

#inverse function theorem #Jacobian #local invertibility

Partial derivatives

Ex.1799Jacobian of a composition

#Jacobian #composition #chain rule

Partial derivatives

Ex.1800Level curves and gradients

#level curves #gradient #orthogonality

Partial derivatives

Ex.1801Mean value inequality

#mean value #inequality #gradient

Partial derivatives

Ex.1802Taylor expansion of order 2

#Taylor expansion #multivariable #approximation

Partial derivatives

Ex.1803Difference exactly $10$

#pigeonhole #difference #exact value

Pigeonhole principle

Ex.1804Difference of at most $10$

#pigeonhole #difference #bound

Pigeonhole principle

Ex.1805Circles and pigeonhole

#pigeonhole #circles #intersection

Pigeonhole principle

Ex.1806Coloring integers

#pigeonhole #coloring #multiples

Pigeonhole principle

Ex.1807Friends and enemies

#pigeonhole #Ramsey #relation

Pigeonhole principle

Ex.1808Graph theory via pigeonhole

#pigeonhole #graph #degree

Pigeonhole principle

Ex.1809Increasing or decreasing subsequence

#pigeonhole #Erdős–Szekeres #subsequence

Pigeonhole principle

Ex.1810Monochromatic triangle

#pigeonhole #Ramsey #monochromatic

Pigeonhole principle

Ex.1811Rectangle with lattice points

#pigeonhole #lattice #rectangle

Pigeonhole principle

Ex.1812Separated integers

#pigeonhole #distance #bounded

Pigeonhole principle

Ex.1813Subsets of fixed size

#pigeonhole #subsets #size

Pigeonhole principle

Ex.1814A subset with sum divisible by $n$

#pigeonhole #subset sum #divisibility

Pigeonhole principle

Ex.1815Sum of two distinct elements

#pigeonhole #sum #distinct

Pigeonhole principle

Ex.1816A triple summing to a multiple

#pigeonhole #triple #multiple

Pigeonhole principle

Ex.1817Triangle area via pigeonhole

#pigeonhole #triangle #area

Pigeonhole principle

Ex.1818Bézout's identity for polynomials

#Bézout #polynomials #linear combination

Ex.1819Complex factorization via roots of unity

#roots of unity #factorization #complex

Ex.1820Decomposition into irreducible factors

#factorization #irreducible factors #real polynomials

Ex.1821A polynomial identity

#polynomial identity #factorization #symmetric

Ex.1822GCD of two polynomials

#GCD #Euclidean algorithm #polynomials

Ex.1823Lagrange interpolation

#interpolation #Lagrange #polynomial

Ex.1824Irreducibility of a polynomial

#irreducibility #polynomials #roots

Ex.1825Roots and coefficients

#roots #coefficients #bounds

Ex.1826A polynomial with integer coefficients

#integer coefficients #roots #Gauss lemma

Ex.1827A polynomial with parameter

#parameter #roots #Vieta

Ex.1828Rolle's theorem for polynomials

#Rolle #roots #derivatives

Ex.1829Roots of a cubic and discriminant

#cubic #discriminant #roots

Ex.1830Symmetric functions of roots

#symmetric functions #Vieta #Newton identities

Ex.1831Bertrand's postulate (statement and examples)

#Bertrand #primes #interval

Ex.1832Quadratic reciprocity applied

#quadratic reciprocity #Legendre #application

Ex.1833Primes of the form $4n+1$

#Dirichlet #arithmetic progression #primes

Ex.1834Fermat test detects compositeness

#Fermat test #witness #compositeness

Ex.1835Fermat factorization

#Fermat factorization #factoring #difference of squares

Ex.1836GCD via Euler's totient

#gcd #Euler totient #formula

Ex.1837Primorial and GCD

#primorial #gcd #primes

Ex.1838Bounds on the $n$-th prime

#n-th prime #bounds #elementary

Ex.1839Prime numbers and prime ideals

#prime ideal #Z #prime numbers

Ex.1840Finding primitive roots

#primitive root #mod prime #computation

Ex.1841Pseudoprimes

#pseudoprime #Fermat #compositeness

Ex.1842Primes and polynomial irreducibility

#irreducibility #primes #polynomial

Ex.1843The sieve of Sundaram

#sieve #Sundaram #primes

Ex.1844Canonical form of a quadratic form

#canonical form #Gauss reduction #signature

Quadratic forms

Ex.1845Change of variables in a quadratic form

#change of variables #quadratic form #matrix

Quadratic forms

Ex.1846Comparison of two quadratic forms

#comparison #positive definite #inequalities

Quadratic forms

Ex.1847The cone $x^2+y^2-z^2=0$

#cone #quadratic form #signature

Quadratic forms

Ex.1848Diagonalization by congruence

#congruence #diagonalization #quadratic form

Quadratic forms

Ex.1849Reduction of a quadratic form by eigenvalues

#eigenvalues #orthogonal diagonalization #quadratic form

Quadratic forms

Ex.1850The ellipse $x^2+xy+y^2=1$

#ellipse #quadratic form #geometry

Quadratic forms

Ex.1851Gram matrix of a quadratic form

#Gram matrix #quadratic form #polar form

Quadratic forms

Ex.1852Checking positivity of a quadratic form

#positive definite #positivity #quadratic form

Quadratic forms

Ex.1853Positive definiteness of a 2×2 parameter family

#parameter #positive definite #minors

Quadratic forms

Ex.1854Reduction of a quadratic form with cross terms

#reduction #completing square #cross terms

Quadratic forms

Ex.1855Signature of a quadratic form in 3 variables

#signature #reduction #quadratic form

Quadratic forms

Ex.1856Sylvester's criterion for positive definiteness

#Sylvester criterion #positive definite #minors

Quadratic forms

Ex.1857Alternation depth of quantifiers

#quantifier alternation #depth #complexity

Quantifiers and negation

Ex.1858Scope of quantifiers

#scope #quantifiers #ambiguity

Quantifiers and negation

Ex.1859When distribution fails

#distribution #counterexample #quantifier

Quantifiers and negation

Ex.1860Proving a quantified statement with $\exists\forall$

#exists forall #nested #olympiad

Quantifiers and negation

Ex.1861Logical equivalence of quantified statements

#equivalence #quantifiers #distribution

Quantifiers and negation

Ex.1862Negating a complex statement

#negation #practice #nested

Quantifiers and negation

Ex.1863Prenex normal form

#prenex form #quantifiers #normal form

Quantifiers and negation

Ex.1864A statement with a specific structure

#epsilon #forall #exists

Quantifiers and negation

Ex.1865Algebraic numbers

#algebraic numbers #minimal polynomial #degree

Rings and fields

Ex.1866Boolean rings

#Boolean ring #idempotent #commutative

Rings and fields

Ex.1867Chinese remainder theorem for rings

#Chinese remainder #coprime ideals #isomorphism

Rings and fields

Ex.1868Eisenstein's criterion

#Eisenstein criterion #irreducibility #polynomials

Rings and fields

Ex.1869$\mathbb{Z}[i]$ is Euclidean

#Euclidean domain #Gaussian integers #division

Rings and fields

Ex.1870Field extensions

#field extension #degree #minimal polynomial

Rings and fields

Ex.1871Finite fields

#finite field #order #prime power

Rings and fields

Ex.1872The Gaussian integers $\mathbb{Z}[i]$

#Gaussian integers #ring #norm

Rings and fields

Ex.1873GCD in $\mathbb{R}[X]$

#GCD #Euclidean algorithm #Bézout

Rings and fields

Ex.1874Irreducibility in $\mathbb{R}[X]$ and $\mathbb{C}[X]$

#irreducibility #polynomial #fields

Rings and fields

Ex.1875Maximal ideals in $\mathbb{R}[X]$

#maximal ideals #polynomials #irreducibility

Rings and fields

Ex.1876Nilpotent elements

#nilpotent #ring #quotient

Rings and fields

Ex.1877Principal ideal domains

#PID #principal ideal #examples

Rings and fields

Ex.1878The field $\mathbb{R}[X]/(X^2+1)$

#quotient ring #polynomial #complex numbers

Rings and fields

Ex.1879Prime characteristic

#characteristic #prime #field

Rings and fields

Ex.1880Prime and maximal ideals

#prime ideal #maximal ideal #quotient

Rings and fields

Ex.1881Quotient rings

#quotient ring #ideal #operations

Rings and fields

Ex.1882Russell's paradox

#Russell paradox #set theory #self-reference

Ex.1883Asymptotic expansion of (1 + 1/n)ⁿ

#asymptotic expansion #sequences #composition of expansions

Taylor expansions

Ex.1884Accumulation points of $\mathbb{Q}$

#accumulation #rationals #density

Topology of ℝ

Ex.1885The Cantor set is compact

#Cantor set #compactness #closed

Topology of ℝ

Ex.1886Compact sets in $\mathbb{R}$

#compact sets #Heine-Borel #closed bounded

Topology of ℝ

Ex.1887Sequential compactness in $\mathbb{R}$

#sequential compactness #Bolzano-Weierstrass

Topology of ℝ

Ex.1888Continuity via open preimages

#continuity #open preimages #topological definition

Topology of ℝ

Ex.1889Limit superior and limit inferior

#limit superior #limit inferior #sequences

Topology of ℝ

Ex.1890The Borel–Lebesgue property

#compactness #open cover #finite subcover

Topology of ℝ

Ex.1891Baire category theorem for $\mathbb{R}^n$

#Baire #complete metric #category

Topology of ℝⁿ

Ex.1892Continuity and open preimages

#continuity #open preimages #multivariable

Topology of ℝⁿ

Ex.1893Ball and cube are homeomorphic

#homeomorphism #ball #cube

Topology of ℝⁿ

Ex.1894Isometries of $\mathbb{R}^n$

#isometry #Euclidean #rigid motion

Topology of ℝⁿ

Ex.1895Heine–Cantor in $\mathbb{R}^n$

#uniform continuity #Heine-Cantor #compactness

Topology of ℝⁿ

Ex.1896Completing a family into a basis

#basis #completion #incomplete basis theorem

Vector spaces and subspaces

Ex.1897Change of basis in a polynomial space

#change of basis #transition matrix #polynomials

Vector spaces and subspaces

Ex.1898Existence of a supplementary subspace

#supplementary subspace #basis extension #existence

Vector spaces and subspaces

Ex.1899Hyperplanes and linear forms

#hyperplane #linear form #kernel #dimension

Vector spaces and subspaces

Ex.1900Dimension of a quotient space

#quotient space #dimension #proof

Vector spaces and subspaces

Ex.1901Direct sum of subspaces

#direct sum #supplementary subspaces #dimension

Vector spaces and subspaces

Ex.1902Decomposing a vector in a direct sum

#direct sum #decomposition #projection

Vector spaces and subspaces

Ex.1903Gauss divergence theorem for a cylinder

#divergence #cylinder #flux

Vector calculus

Ex.1904The divergence theorem

#divergence theorem #flux #surface

Vector calculus

Ex.1905Flux through a cube

#divergence #cube #flux

Vector calculus

Ex.1906The dual basis

#dual space #dual basis #linear forms

Vector spaces and subspaces

Ex.1907Eigenspaces sum to the whole space

#eigenspaces #direct sum #diagonalization

Vector spaces and subspaces

Ex.1908Divergence theorem with variable divergence

#divergence #sphere #variable density

Vector calculus

Ex.1909Orthonormal basis of a subspace

#Gram-Schmidt #orthonormal basis #subspace

Vector spaces and subspaces

Ex.1910Recovering a field from its curl (2D)

#Green #curl #reconstruction

Vector calculus

Ex.1911Green's reciprocity

#Green #reciprocity #symmetry

Vector calculus

Ex.1912Green's theorem on an annulus

#Green #annulus #inner boundary

Vector calculus

Ex.1913Green's theorem

#Green theorem #curl #region

Vector calculus

Ex.1914Green's theorem with a general field

#Green #general field #region

Vector calculus

Ex.1915Intersection of two kernels

#intersection #kernel #dimension

Vector spaces and subspaces

Ex.1916Intersection and sum of subspaces

#intersection #sum #dimension formula

Vector spaces and subspaces

Ex.1917Area of a region using line integral

#line integral #area #polar

Vector calculus

Ex.1918Conservative fields in 3D

#conservative #3D #potential

Vector calculus

Ex.1919Finding a potential function

#potential #conservative #integration

Vector calculus

Ex.1920Matrix of a linear map in a given basis

#matrix representation #basis #linear map

Vector spaces and subspaces

Ex.1921Rank-nullity on a 3×4 matrix

#rank-nullity #rank #matrix

Vector spaces and subspaces

Ex.1922Orthogonal complement

#orthogonal complement #dimension #inner product

Vector spaces and subspaces

Ex.1923A subspace of polynomials with a root condition

#polynomial subspace #evaluation #dimension

Vector spaces and subspaces

Ex.1924Quotient vector space

#quotient space #cosets #dimension

Vector spaces and subspaces

Ex.1925Stokes with a hemisphere

#Stokes #hemisphere #closed curve

Vector calculus

Ex.1926Stokes' theorem

#Stokes theorem #curl #surface

Vector calculus

Ex.1927A subspace depending on a parameter

#parameter #subspace #dimension

Vector spaces and subspaces

Ex.1928Sum of two planes in $\mathbb{R}^3$

#sum #intersection #planes

Vector spaces and subspaces

Ex.1929Symmetric and skew-symmetric matrices

#symmetric #skew-symmetric #direct sum

Vector spaces and subspaces

Ex.1930The subspace of traceless matrices

#trace #subspace #dimension #matrices

Vector spaces and subspaces

Ex.1931Kernel of the transpose

#transpose #kernel #orthogonal complement #rank

Vector spaces and subspaces

Ex.1932Binomial coefficients in steps of three

#roots of unity filter #binomial coefficients #complex numbers

Complex numbers

Ex.1933Irreducibility with Eisenstein's criterion

#Eisenstein #irreducibility #cyclotomic polynomial #shift trick

Ex.1934A polynomial that equals k/(k+1) at 0, 1, …, n

#polynomial interpolation #degree argument #Lagrange

Ex.1935A polynomial with all roots real and Vieta

#Vieta #discriminant #real roots #Newton inequality

Ex.1936Polynomials with P(x²) = P(x)²

#polynomial equation #degree argument #coefficients comparison

Ex.1937The product of sin(kπ/n)

#roots of unity #trigonometric product #cyclotomic

Complex numbers

Ex.1938Triangle-free graphs have at most n²/4 edges (Mantel)

#double counting #graphs #Cauchy–Schwarz #extremal graph theory

Ex.1939Paths that never go below the axis: Catalan numbers

#reflection principle #Catalan numbers #lattice paths #bijection

Binomial coefficients

Ex.1940Graphs with no 4-cycle

#extremal graph theory #double counting #convexity #cycles

Ex.1941Hall's marriage theorem and regular bipartite graphs

#Hall's theorem #matchings #bipartite graphs #induction

Ex.1942The ballot problem

#ballot problem #reflection principle #lattice paths #probability

Counting principles

Ex.1943An inequality as a functional equation (IMO 1977)

#IMO #natural numbers #induction #monotonicity

Functional equations

Ex.1944The regular heptagon identity

#Ptolemy's theorem #regular heptagon #chords

Ex.1945Squares on two sides and the median

#complex numbers #squares #median #perpendicularity

Ex.1946The Euler line, by vectors

#Euler line #orthocentre #centroid #circumcentre #vectors

Ex.1947The three external bisectors meet the sides on a line

#Menelaus's theorem #angle bisector theorem #collinearity

Ex.1948Three equal products at the orthocentre

#radical axis #radical centre #orthocentre #circles on diameters

Ex.1949The one-seventh triangle

#area ratios #cevians #affine invariance #coordinates

Ex.1950A classic with abc = 1 (IMO 1995)

#IMO #substitution #Titu's lemma #abc=1

Ex.1951Mixing variables: a²b + b²c + c²a ≤ 4/27

#mixing variables #cyclic sums #ordering #AM-GM

Ex.1952Schur's inequality and a bound with a + b + c = 1

#Schur's inequality #IMO 1984 #symmetric polynomials #ordering variables

Ex.1953Cauchy's extended mean value theorem

#Cauchy #extended MVT #ratio

Mean Value Theorem

Ex.1954Darboux's theorem on derivatives

#Darboux #intermediate value #derivatives

Mean Value Theorem

Ex.1955Second-order Taylor formula

#Taylor #MVT #second order

Mean Value Theorem

Ex.1956Iterated Rolle

#Rolle #higher derivatives #iterated

Mean Value Theorem

Ex.1957⌊√n + √(n+1)⌋ = ⌊√(4n+2)⌋

#floor function #square roots #squares modulo 4

Divisibility and gcd

Ex.1958Primes 3 mod 4 and sums of two squares

#sum of two squares #quadratic residues #descent

Ex.1959When is n⁴ + n³ + n² + n + 1 a perfect square?

#bounding between squares #Diophantine equations

Diophantine equations

Ex.1960An integrality proof by Legendre's formula

#Legendre's formula #floor function #integrality

Ex.1961Prime factors of Fermat numbers

#order #Fermat numbers #Fermat's little theorem

Divisibility and gcd

Ex.1962The exact power of 3 dividing 2^(3ⁿ) + 1

#p-adic valuation #lifting the exponent #induction

Ex.1963Classical closed subgroups of $GL_n(\mathbb{R})$

#closed subgroups #GL_n #SL_n #O_n #determinant

Topology of Mn(ℝ)

Ex.1964Compactness of orthogonal and orthogonal groups

#compactness #O_n #SO_n #boundedness

Topology of Mn(ℝ)

Ex.1965Continuity of eigenvalues (statement)

#eigenvalues #continuity #roots of polynomial

Topology of Mn(ℝ)

Ex.1966A homeomorphism of $M_n(\mathbb{R})$

#homeomorphism #matrix map #inverse map

Topology of Mn(ℝ)

Ex.1967Continuity of the matrix exponential

#matrix exponential #continuity #series

Topology of Mn(ℝ)

Ex.1968$GL_n(\mathbb{R})$ is open in $M_n(\mathbb{R})$

#GL_n #open set #determinant continuity

Topology of Mn(ℝ)

Ex.1969P(x)P(x − 1) = P(x²)

#polynomial equation #roots #unit circle #functional equation

Ex.1970gcd(F_m, F_n) = F_gcd(m,n)

#Fibonacci numbers #gcd #Euclidean algorithm #identities

Equations and radicals

Ex.1971A polynomial that equals 1 / C(n+1, k) at k = 0, …, n

#polynomials #interpolation #finite differences #binomial coefficients

Ex.1972Chomp and the strategy-stealing argument

#combinatorial games #strategy stealing #Chomp #invariant

Counting principles

Ex.1973Erdős's lower bound for Ramsey numbers

#probabilistic method #Ramsey theory #expectation

Ex.1974Hall's marriage theorem and a deck of cards

#Hall's theorem #matchings #bipartite graphs #induction

#linear algebra method #parity #set systems #F2

Counting principles

Ex.1976Sperner's theorem and the LYM inequality

#antichains #double counting #chains #Sperner

Binomial coefficients

Ex.1977The Sylvester–Gallai theorem

#combinatorial geometry #extremal principle #minimal distance

Pigeonhole principle

Ex.1978Turán's theorem

#graph theory #extremal #induction #Turán

Ex.1979Additive functions with f(x²) = f(x)²

#additive function #monotonicity #Cauchy equation

Functional equations

Ex.1980Solving f(f(n)) = n + 2 on the positive integers

#functional equation #integers #parity #recursion

Functional equations

Ex.1981f(f(x) + y) = 2x + f(f(y) − x)

#surjectivity #substitution #linear functions

Functional equations

Ex.1982IMO 1968: f(x + a) = ½ + √(f(x) − f(x)²)

#IMO 1968 #periodicity #squaring #completing squares

Functional equations

Ex.1983IMO 2010: f(⌊x⌋y) = f(x)⌊f(y)⌋

#IMO 2010 #floor function #functional equation #case analysis

Functional equations

Ex.1984Euler's formula OI² = R² − 2Rr

#Euler's formula #incentre #circumcircle #power of a point #incentre-excentre lemma

Ex.1985Harmonic division from Ceva and Menelaus

#Ceva #Menelaus #harmonic conjugates #cevians

Ex.1986Ptolemy's inequality by inversion

#inversion #Ptolemy's inequality #triangle inequality

Ex.1987The butterfly theorem

#butterfly theorem #chords #coordinates #Vieta

Analytic geometry

Ex.1988The symmedian and the tangents at B and C

#symmedian #tangent-chord angle #areas #law of sines

Ex.1989Trigonometric Ceva and a hidden isosceles triangle

#trigonometric Ceva #angle chasing #monotonicity

Ex.1990A cyclic inequality: a²b + b²c + c²a + abc ≤ 4

#cyclic inequality #WLOG ordering #AM-GM

Ex.1991With abc = 1: Σ 1/(a² + a + 1) ≥ 1

#constraint abc=1 #clearing denominators #expansion

Ex.1992Hölder's inequality for three sequences

#Hölder #AM-GM #normalisation

Ex.1993Two inequalities by induction on the number of variables

#induction #AM-GM #Bernoulli #smoothing

Ex.1994The cube substitution: Σ 1/(a + b + 1) ≤ 1

#substitution #abc=1 #cubes #AM-GM

Ex.1995Even perfect numbers (Euclid–Euler)

#perfect numbers #sigma function #multiplicativity #Mersenne primes

Divisibility and gcd

Ex.1996Every prime ≡ 1 (mod 4) is a sum of two squares

#sum of two squares #Thue's lemma #pigeonhole #Wilson

Ex.1997n divides φ(aⁿ − 1)

#order #Euler's theorem #totient

Ex.1998Which primes divide C(2n, n)?

#Legendre's formula #central binomial coefficient #Bertrand's postulate

Ex.1999gcd(F_m, F_n) = F_gcd(m,n)

#Fibonacci numbers #gcd #Euclidean algorithm #divisibility sequences

Divisibility and gcd

Ex.2000A²+B² is singular when A³ = B³ and A²B = B²A

#kernel #algebraic identity #Putnam

Applications and the case n = 2

Ex.2001The determinant of a block matrix with commuting lower blocks

#density #block matrices #Schur complement

Applications and the case n = 2

Ex.2002Invertible skew-symmetric matrices have two components

#Pfaffian #connected components #skew-symmetric matrices

Applications and the case n = 2

Ex.2003Rotations by Ï€ form a projective plane; the Cayley transform covers the rest

#SO(3) #Cayley transform #projective plane #cut locus

Applications and the case n = 2

Ex.2004adj(adj A) = (det A)^(n-2) A

#adjugate #density #polynomial identity

Applications and the case n = 2

Ex.2005det(A² + B²) is positive for commuting real matrices

#complexification #determinant #commuting matrices

Applications and the case n = 2

Ex.2006Ky Fan's maximum principle

#Ky Fan #eigenvalues #orthogonal projections #convexity

Applications and the case n = 2

Ex.2007M_n(R)/(O(n) . O(n)) is the space of singular values

#singular values #SVD #orbit space #Weyl inequality

Applications and the case n = 2

Ex.2008M_n(R)/O(n) is the cone of positive semidefinite matrices

#quotient space #proper maps #positive semidefinite #orbit space

Applications and the case n = 2

Ex.2009Symmetric matrices with distinct eigenvalues form a connected set

#connectedness #symmetric matrices #spectral theorem #SO(n)

Applications and the case n = 2

Ex.2010Orbits of real-diagonalisable matrices are closed

#conjugacy orbit #minimal polynomial #closed sets

Density and closures

Ex.2011Orbits of non-diagonalisable matrices are not closed

#conjugacy orbit #Jordan form #scaling #closure

Density and closures

Ex.2012The regular nilpotent orbit is dense in the nilpotent cone

#nilpotent matrices #Jordan form #orbit closure

Density and closures

Ex.2013A subgroup of SO(4) that is not closed

#irrational winding #torus #Kronecker #non-closed subgroup

Exponential, logarithm and group topology

Ex.2014Compact subgroups of SL_n are conjugate into SO(n)

#compact group #Haar measure #invariant inner product #conjugation

Exponential, logarithm and group topology

Ex.2015Every continuous one-parameter subgroup is e^(tA)

#one-parameter subgroup #regularity #integration trick

Exponential, logarithm and group topology

Ex.2016SO(3) is homeomorphic to the 3-sphere

#SO(3) #quaternions #real projective space #double cover

Exponential, logarithm and group topology

Ex.2017A neighbourhood of the identity generates the connected group

#generating set #open subgroup #connected group #GL_n^+

Exponential, logarithm and group topology

Ex.2018The fundamental group of SO(3)

#fundamental group #covering space #SO(3) #deck transformations

Exponential, logarithm and group topology

Ex.2019The Lie algebra of a closed subgroup is a linear subspace

#Lie algebra #closed subgroup #Lie product formula

Exponential, logarithm and group topology

Ex.2020The Lie product formula

#Lie product formula #Trotter #norm estimates #telescoping

Exponential, logarithm and group topology

Ex.2021Which matrices of the special linear group are exponentials?

#SL_2 #trace #Cayley-Hamilton #exponential image

Exponential, logarithm and group topology

Ex.2022A traceless matrix has a zero diagonal in some orthonormal basis

#trace #numerical range #orthonormal basis #induction

Rank, spectrum and norms

Ex.2023The fibres of (trace, determinant) on 2×2 matrices

#conjugacy classes #2x2 matrices #hyperboloid #trace #determinant

Rank, spectrum and norms

Ex.2024Dobrushin's contraction for positive stochastic matrices

#stochastic matrices #Dobrushin #oscillation #convergence of powers

Rank, spectrum and norms

Ex.2025Eigenvalues of normal matrices move by at most the size of the perturbation

#normal matrices #Bauer-Fike #Weyl #unitary diagonalisation

Rank, spectrum and norms

Ex.2026A stochastic matrix has a stationary distribution

#stochastic matrices #compactness #Cesàro averages #Markov chains

Rank, spectrum and norms

Ex.2027Gelfand's spectral radius formula

#spectral radius #Gelfand formula #norms #powers

Rank, spectrum and norms

Ex.2028Lyapunov's criterion for Hurwitz matrices

#Hurwitz #Lyapunov equation #stability #positive definite

Rank, spectrum and norms

Ex.2029The spectral radius is monotone on nonnegative matrices

#spectral radius #nonnegative matrices #Gelfand formula #monotonicity

Rank, spectrum and norms

Ex.2030Sylvester's criterion and the openness of the positive cone

#Sylvester criterion #positive definite #LDL decomposition #leading minors

Rank, spectrum and norms

Ex.2031The Bauer-Fike theorem

#Bauer-Fike #perturbation of eigenvalues #condition number #diagonalisable

Rank, spectrum and norms

Ex.2032The Eckart-Young theorem in the operator norm

#Eckart-Young #low rank approximation #singular values #dimension counting

Rank, spectrum and norms

Ex.2033Matrices with two distinct real eigenvalues retract onto a circle

#homotopy equivalence #deformation retraction #2x2 matrices #discriminant

Rank, spectrum and norms

Ex.2034Four series involving logarithms

#root test #comparison #equivalent #logarithm

Convergence tests

Ex.2035Hadamard's inequality for determinants

#Hadamard inequality #QR decomposition #Gram-Schmidt #determinant

Norms and metric structure

Ex.2036An upper bound of the determinant

#Hadamard inequality #AM-GM #equality case #determinant

Norms and metric structure

Ex.2037The determinant is Lipschitz on bounded sets

#determinant #Lipschitz #telescoping #Hadamard inequality

Norms and metric structure

Ex.2038The extreme points of the operator-norm unit ball

#extreme points #orthogonal group #singular value decomposition #convexity

Norms and metric structure

Ex.2039The operator-norm ball is the convex hull of O(n)

#convex hull #orthogonal group #SVD #Krein-Milman

Norms and metric structure

Ex.2040A majoration of the spectral radius

#spectral radius #Gelfand formula #symmetric matrices #powers

Norms and metric structure

Ex.2041The set {Ï(A) < 1}: open, with boundary {Ï = 1}

#spectral radius #continuity of eigenvalues #closure #boundary

Open, closed, interior, boundary

Ex.2042The interior of the real-diagonalisable matrices

#diagonalisable #interior #distinct eigenvalues #Jordan blocks

Open, closed, interior, boundary

Ex.2043The closure of the real-diagonalisable matrices

#diagonalisable #closure #triangularisation #continuity of roots

Open, closed, interior, boundary

Ex.2044Matrices with a repeated eigenvalue are rare

#discriminant #algebraic sets #polynomials #nowhere dense

Open, closed, interior, boundary

Ex.2045Symmetric matrices with distinct eigenvalues are open and dense

#symmetric matrices #spectral theorem #genericity #distinct eigenvalues

Open, closed, interior, boundary

Ex.2046Weyl's inequality for singular values

#singular values #Courant-Fischer #perturbation #Lipschitz

Continuity of matrix maps

Ex.2047Weyl's inequality for eigenvalues of symmetric matrices

#eigenvalues #symmetric matrices #Courant-Fischer #perturbation

Continuity of matrix maps

Ex.2048The PSD square root is continuous

#matrix square root #compactness #uniqueness #PSD cone

Continuity of matrix maps

Ex.2049Polar decomposition is a homeomorphism

#polar decomposition #homeomorphism #orthogonal group #positive definite

Continuity of matrix maps

Ex.2050GL_n is homeomorphic to O(n) × upper triangular

#QR decomposition #Gram-Schmidt #Iwasawa #homeomorphism

Continuity of matrix maps

Ex.2051Cholesky: a homeomorphism from T_+ onto PD

#Cholesky decomposition #positive definite matrices #QR #homeomorphism

Continuity of matrix maps

Ex.2052The orthogonal matrix nearest to a given matrix

#orthogonal Procrustes #SVD #Frobenius norm #polar decomposition

Continuity of matrix maps

Ex.2053The Cayley transform of a skew-symmetric matrix

#Cayley transform #skew-symmetric #special orthogonal group #chart

Continuity of matrix maps

Ex.2054No continuous choice of a unit eigenvector

#eigenvectors #Möbius band #connectedness #no continuous section

Continuity of matrix maps

Ex.2055SO(2) has no continuous square root

#path lifting #SO(2) #square root #covering space

Continuity of matrix maps

Ex.2056Complex structures on the set A² = -I

#complex structure #A squared = -I #2x2 matrices #homeomorphism

Ex.2057Connectedness of conjugacy orbits

#conjugacy orbits #centraliser #group action #rotation matrix

Ex.2058SO(n) is a deformation retract of the general linear group

#deformation retraction #homotopy equivalence #polar decomposition

Ex.2059Matrices with n distinct real eigenvalues

#diagonalisable matrices #Weyl chamber #conjugation #path connected

Ex.2060Components of the invertible symmetric matrices

#symmetric matrices #signature #eigenvalues #Weyl inequality

Ex.2061GL(2,ℝ) is not simply connected

#fundamental group #deformation retraction #circle #homotopy

Ex.20622×2 matrices without real eigenvalues

#2x2 matrices #discriminant #complex eigenvalues #components

Ex.2063Ordered real eigenvalues are continuous on the real-rooted matrices

#continuity of roots #ordered eigenvalues #Rouché #real-rooted matrices

Applications and the case n = 2

Ex.2064The complex structures A² =-I on form exactly two components

#conjugacy orbits #complex structures #orientation #connected components #GL_n

Applications and the case n = 2

Ex.2065The operator-norm unit sphere of M_2(R) is a 3-sphere

#operator norm #unit sphere #M_2(R) #radial projection #complex coordinates

Applications and the case n = 2

Ex.2066The orbit of a regular nilpotent matrix is connected iff n is odd

#nilpotent matrices #conjugacy orbits #cyclic vectors #connected components

Applications and the case n = 2

Ex.2067Tangent of sine minus sine of tangent

#Taylor expansion #composition #high order #challenge

Ex.2068An absolute value integral with a parameter

#absolute value #parameters #trigonometry

Riemann integral

Ex.2069The root of x^n + x = 1

#implicit sequences #monotonicity #intermediate value theorem

Ex.2070Rayleigh quotient and eigenvalues

#Rayleigh quotient #eigenvalues #optimisation on the sphere

Optimisation in several variables

Ex.2071The AM-GM inequality via optimisation

#inequality #AM-GM #optimisation #proof

Optimisation in several variables

Ex.2072Convergence of the series of sin(n)/n

#Abel summation #Dirichlet test #trigonometric series

Convergence tests

Ex.2073Interlacing an arithmetic and a geometric progression

#arithmetic progression #geometric progression #inequalities #construction

Ex.2074Rearranging a conditionally convergent series

#conditional convergence #rearrangement #Riemann theorem #ln 2

Ex.2075Legendre polynomials have all their roots in (-1, 1)

#Rolle's theorem #polynomials #induction #Legendre

Mean Value Theorem

Ex.2076Superposition of right-hand sides

#superposition #double root #polynomial right-hand side

Linear equations of order two

Ex.2077The integral of sin x / x

#improper integrals #conditional convergence #integration by parts

Improper integrals

Ex.2078An integral that equals its own mirror image

#symmetry #substitution #logarithm #tangent

Integration techniques

Ex.2079The logistic sequence decays like 1/n

#Cesaro lemma #asymptotic equivalent #recurrence

Ex.2080The popcorn function

#continuity #rationals #density #pathological functions

Continuous and discontinuous functions

Ex.2081Reciprocals of integers with no digit 9

#comparison test #counting #harmonic series #Kempner

Convergence tests

Ex.2082A series with an equivalent that converges but does not

#equivalents #alternating series #counterexample

Convergence tests

Ex.2083A smooth function that is not its Taylor series

#flat function #smooth #analytic #counterexample

Taylor expansions

Ex.2084Chu–Vandermonde for negative upper index

#Chu-Vandermonde #generalized binomial #identity

Binomial coefficients

Ex.2085A hard combinatorial identity

#hard identity #combinatorial #binomial

Binomial coefficients

Ex.2086A correct challenge identity

#challenge #identity #Franel

Binomial coefficients

Ex.2087Gauss's binomial theorem

#Gauss binomial #hypergeometric #identity

Binomial coefficients

Ex.2088Generating function of central binomials

#generating function #central binomial #asymptotics

Binomial coefficients

Ex.2089A challenging binomial sum

#binomial #sum #generating function

Binomial coefficients

Ex.2090A challenge identity

#challenge #identity #binomial

Binomial coefficients

Ex.2091Lucas's theorem

#Lucas #mod p #binomial

Binomial coefficients

Ex.2092Mahonian numbers

#Mahonian #inversion #binomial

Binomial coefficients

Ex.2093Partial fractions with binomials

#partial fraction #binomial #generating function

Binomial coefficients

Ex.2094Riordan arrays (statement)

#Riordan array #generating function #binomial

Binomial coefficients

Ex.2095Zeilberger's algorithm (statement)

#Zeilberger #creative telescoping #binomial

Binomial coefficients

Ex.2096Baire category theorem (statement)

#Baire #category #uncountable

Countability and cardinality

Ex.2097Cantor function

#Cantor function #cardinality #devil staircase

Countability and cardinality

Ex.2098Trichotomy of cardinals (statement)

#trichotomy #cardinals #choice

Countability and cardinality

Ex.2099The continuum hypothesis (statement)

#continuum hypothesis #aleph #independence

Countability and cardinality

Ex.2100The generalized continuum hypothesis (statement)

#GCH #generalized continuum #set theory

Countability and cardinality

Ex.2101Infinite subsets of $\mathbb{N}$

#infinite subsets #uncountable #power set

Countability and cardinality

Ex.2102Proof of Cantor–Bernstein–Schröder

#Cantor-Bernstein #proof #bijection

Countability and cardinality

Ex.2103Solving a trigonometric equation via cubics

#trigonometric equation #cubic #cos

Complex numbers

Ex.2104The Julia set of $z\mapsto z^2$

#Julia set #iteration #complex dynamics

Complex numbers

Ex.2105The Möbius transformation

#Möbius #complex #transformation

Complex numbers

Ex.2106A product formula for sines

#product formula #sine #roots of unity

Complex numbers

Ex.2107Integrating over the unit circle

#unit circle #parametrization #integration

Complex numbers

Ex.2108A Carmichael number check

#Carmichael #Fermat #pseudoprime

Ex.2109The Carmichael function $\lambda(n)$

#Carmichael function #lcm #exponent

Ex.2110The discrete logarithm

#discrete logarithm #cyclic group #Pohlig-Hellman

Ex.2111Power tower modulo $n$

#power tower #modular exponentiation #Carmichael

Ex.2112Square roots modulo $p$

#square roots #mod p #Tonelli-Shanks

Ex.2113Stirling numbers via modular arithmetic

#Stirling #modular arithmetic #combinatorics

Ex.2114NTT and modular arithmetic (statement)

#NTT #number theory transform #FFT #modular

Ex.2115Two equality constraints

#Lagrange #multiple constraints #KKT

Optimisation in several variables

Ex.2116Bertrand's ballot problem

#ballot problem #lattice paths #Catalan

Counting principles

Ex.2117Catalan numbers

#Catalan numbers #paths #Dyck paths

Counting principles

Ex.2118Gram determinant and Cauchy-Schwarz

#Gram determinant #Cauchy-Schwarz #inner product

Ex.2119The general Vandermonde determinant

#Vandermonde #determinant #general formula

Ex.2120Hadamard's inequality

#Hadamard inequality #determinant #column norms

Ex.2121Sylvester's determinant identity

#Sylvester identity #block matrices #determinant

Ex.2122Carmichael numbers (statement)

#Carmichael #primes #Fermat

Divisibility and gcd

Ex.2123The Miller–Rabin test

#primality testing #Miller-Rabin #modular arithmetic

Divisibility and gcd

Ex.2124The Cayley-Hamilton theorem

#Cayley-Hamilton #characteristic polynomial #matrix identity

Eigenvalues and diagonalisation

Ex.2125Eigenvalues of a circulant matrix

#circulant matrix #eigenvalues #roots of unity

Eigenvalues and diagonalisation

Ex.2126Powers of a companion matrix and Fibonacci numbers

#companion matrix #Fibonacci #diagonalization

Eigenvalues and diagonalisation

Ex.2127Diagonalizability depending on a parameter

#diagonalizability #parameter #eigenspaces

Eigenvalues and diagonalisation

Ex.2128Trigonalization and eigenvalues

#trigonalization #characteristic polynomial #real eigenvalues

Eigenvalues and diagonalisation

Ex.2129Diagonalizability and the minimal polynomial

#minimal polynomial #diagonalization #criterion

Eigenvalues and diagonalisation

Ex.2130Spectral decomposition of a symmetric matrix

#spectral decomposition #symmetric matrix #projection

Eigenvalues and diagonalisation

Ex.2131Orthogonal diagonalization of a symmetric matrix

#symmetric matrix #orthogonal diagonalization #spectral theorem

Eigenvalues and diagonalisation

Ex.2132A radical equation with parameter (advanced)

#parameter #radical equation #discussion

Equations and radicals

Ex.2133A nonlinear 3×3 system

#nonlinear system #olympiad #symmetry

Equations and radicals

Ex.2134A cyclic system

#cyclic system #symmetric #olympiad

Equations and radicals

Ex.2135A symmetric 3-variable system

#symmetric system #Vieta #olympiad

Equations and radicals

Ex.2136Edge coloring (Vizing)

#Vizing #edge coloring #chromatic index

Ex.2137Erdős–Gallai theorem (statement)

#Erdős-Gallai #degree sequence #graphical

Ex.2138Erdős–Gyárfás conjecture (statement)

#Erdős-Gyárfás #conjecture #cycles

Ex.2139Expander graphs (statement)

#expander #spectral gap #connectivity

Ex.2140The four-color theorem proof (history)

#four-color #history #proof

Ex.2141Kuratowski's theorem (statement)

#Kuratowski #planarity #minors

Ex.2142Graph minors

#minor #Robertson-Seymour #structure

Ex.2143Perfect graphs

#perfect graph #clique #chromatic

Ex.2144Spectral graph theory (statement)

#spectral graph #eigenvalues #adjacency

Ex.2145Szemerédi's regularity lemma (statement)

#Szemerédi #regularity #dense graphs

Ex.2146Turán's theorem

#Turán #extremal #clique

Ex.2147Burnside's lemma

#Burnside lemma #group action #counting

Ex.2148The class equation

#class equation #conjugacy #center

Ex.2149Structure of finite abelian groups

#abelian group #structure theorem #decomposition

Ex.2150Free groups (statement)

#free group #generators #relations

Ex.2151Group actions and orbits

#group action #orbit #stabilizer

Ex.2152Groups of order $p^2$

#p-group #abelian #order

Ex.2153Sylow's first theorem

#Sylow #p-subgroup #existence

Ex.2154Sylow's third theorem

#Sylow #congruence #number of subgroups

Ex.2155Both rows and columns

#IE #matrices #both

Inclusion-exclusion

Ex.2156Exactly $k$ properties

#IE #exactly k #sieve

Inclusion-exclusion

Ex.2157The ménage problem (statement)

#ménage #IE #arrangements

Inclusion-exclusion

Ex.2158Cyclic SOS decomposition

#SOS #cyclic #symmetric

Ex.2159Hölder's inequality

#Hölder #inequality #conjugate exponents

Ex.2160A direct application of Hölder

#Hölder #powers #inequalities

Ex.2161Muirhead's inequality

#Muirhead #symmetric inequalities #majorization

Ex.2162Generalized Nesbitt inequality

#Nesbitt #generalization #n variables

Ex.2163Schur's inequality

#Schur #symmetric inequalities #factorization

Ex.2164Sum of squares decomposition

#SOS #sum of squares #factorization

Ex.2165The sum-of-squares method

#SOS #sum of squares #symmetric inequalities

Ex.2166Schur's inequality via SOS

#Schur #SOS #symmetric

Ex.2167The uvw method

#uvw #symmetric inequalities #pqr

Ex.2168Fourier coefficients and orthogonal projection

#Fourier coefficients #trigonometric basis #orthogonal projection

Inner product spaces

Ex.2169Least squares and normal equations

#least squares #normal equations #orthogonal projection

Inner product spaces

Ex.2170Legendre polynomials

#Legendre polynomials #orthogonal polynomials #inner product

Inner product spaces

Ex.2171Orthogonal projection onto a column space

#orthogonal projection #column space #normal equations

Inner product spaces

Ex.2172Riesz representation theorem (finite-dimensional)

#Riesz representation #linear forms #inner product

Inner product spaces

Ex.2173Algebraic independence of $e$ and $\pi$

#algebraic independence #e #pi

Ex.2174Irrationality of e+ π and πe

#irrationality #e #pi #open problem

Ex.2175Euler's number and transcendence

#e #transcendence #Euler

Ex.2176Schanuel's conjecture (statement)

#Schanuel #conjecture #transcendence

Ex.2177Real induction (statement)

#real induction #intervals #completeness

Ex.2178Diagonalizability depending on a parameter

#diagonalization #parameter #characteristic polynomial

Matrices and linear systems

Ex.2179QR factorization by Gram-Schmidt

#QR factorization #Gram-Schmidt #orthogonalization

Matrices and linear systems

Ex.2180Singular value decomposition of a 2×2 matrix

#SVD #singular values #orthogonal matrices

Matrices and linear systems

Ex.2181Orthogonal diagonalization of a symmetric matrix

#symmetric matrix #orthogonal diagonalization #spectral theorem

Matrices and linear systems

Ex.2182Banach fixed point theorem

#Banach #fixed point #contraction

Metric and normed spaces

Ex.2183Gaussian integral over $\mathbb{R}^2$

#Gaussian #polar #improper integral

Multiple integrals

Ex.2184Jacobian of a nonlinear transformation

#Jacobian #nonlinear #change of variables

Multiple integrals

Ex.2185Harmonic conjugate

#harmonic conjugate #Cauchy-Riemann #complex

Partial derivatives

Ex.2186Partial derivatives exist but function is not differentiable

#differentiability #counterexample #partial derivatives

Partial derivatives

Ex.2187Ramsey-type pigeonhole

#pigeonhole #Ramsey #coloring

Pigeonhole principle

Ex.2188Cyclotomic polynomials

#cyclotomic #roots of unity #irreducibility

Ex.2189Common roots of two polynomials

#resultant #common root #gcd

Ex.2190Newton's identities for three variables

#Newton identities #symmetric functions #power sums

Ex.2191The AKS primality test (statement)

#AKS #deterministic primality #polynomial time

Ex.2192Elliptic curve factorization (statement)

#elliptic curve #factorization #Lenstra

Ex.2193The Fermat quotient

#Fermat quotient #prime #congruence

Ex.2194The Lucas–Lehmer test

#Lucas-Lehmer #Mersenne #primality

Ex.2195Chebyshev's bounds

#Chebyshev #bounds #prime counting

Ex.2196Primes in arithmetic progressions

#Dirichlet theorem #arithmetic progression #primes

Ex.2197The prime number theorem (statement)

#prime number theorem #asymptotic #pi function

Ex.2198Existence of primitive roots

#primitive root #existence #cyclic group

Ex.2199Proth's theorem

#Proth #primes #primality

Ex.2200Quadratic reciprocity (statement)

#quadratic reciprocity #Legendre symbol #primes

Ex.2201Sophie Germain's theorem

#Sophie Germain #Fermat #primes

Ex.2202Stirling's formula and primes

#Stirling #factorial #prime counting

Ex.2203The Miller–Rabin test

#Miller-Rabin #primality #probabilistic

Ex.2204Wieferich primes (statement)

#Wieferich #Fermat quotient #primes

Ex.2205Bounding a quadratic form by eigenvalues

#eigenvalues #Rayleigh quotient #inequalities

Quadratic forms

Ex.2206An inequality via a positive definite form

#inequality #positive definite #quadratic form

Quadratic forms

Ex.2207Sylvester's law of inertia

#inertia law #signature #invariance

Quadratic forms

Ex.2208LDL decomposition of a symmetric matrix

#LDL decomposition #Sylvester #quadratic form

Quadratic forms

Ex.2209Maximizing a quadratic form on the sphere

#maximum #sphere #eigenvalues

Quadratic forms

Ex.2210Minimizing a quadratic form on the sphere

#minimum #sphere #eigenvalues

Quadratic forms

Ex.2211Orthogonal diagonalization of a quadratic form

#orthogonal diagonalization #eigenvalues #quadratic form

Quadratic forms

Ex.2212Positive definiteness depending on a parameter

#parameter #positive definite #Sylvester

Quadratic forms

Ex.2213The Rayleigh quotient

#Rayleigh quotient #eigenvalues #optimization

Quadratic forms

Ex.2214Signature depending on a parameter

#signature #parameter #quadratic form

Quadratic forms

Ex.2215When are two quadratic forms equivalent?

#equivalence #signature #congruence

Quadratic forms

Ex.2216Irreducibility of the cyclotomic polynomial

#cyclotomic #irreducibility #field

Rings and fields

Ex.2217Sums of two squares via $\mathbb{Z}[i]$

#sums of squares #Gaussian integers #factorization

Rings and fields

Ex.2218Localization

#localization #fraction field #ring

Rings and fields

Ex.2219The spectrum of a ring (statement)

#spectrum #prime ideals #Zariski topology

Rings and fields

Ex.2220Unique factorization domains

#UFD #unique factorization #examples

Rings and fields

Ex.2221Baire category theorem for $\mathbb{R}$

#Baire #category #dense G-delta

Topology of ℝ

Ex.2222Proving the dimension formula

#dimension formula #proof #intersection and sum

Vector spaces and subspaces

Ex.2223Stokes on a cone

#Stokes #cone #surface

Vector calculus

Ex.2224Stokes on a torus (statement)

#Stokes #torus #closed surface

Vector calculus

Ex.2225Chebyshev polynomials and rational values of cosine

#Chebyshev polynomials #algebraic integers #rational root theorem #Niven

Ex.2226Subsets whose sum is divisible by p

#roots of unity filter #generating functions #counting #modular arithmetic

Counting principles

Ex.2227The Sylvester-Gallai theorem

#combinatorial geometry #extremal principle #distance #lines

Pigeonhole principle

Ex.2228Napoleon's theorem

#complex numbers #rotations #equilateral triangle

Ex.2229The nine-point circle

#nine-point circle #complex numbers #orthocentre #feet of altitudes

Ex.2230The Simson line

#Simson line #complex numbers #collinearity #unit circle

Ex.2231IMO 2001: a/√(a² + 8bc)

#IMO #weighted Jensen #convexity #normalisation

Ex.2232Vasc's inequality by a sum of squares

#SOS #sum of squares #cyclic inequality #Vasc's inequality

Ex.2233Vieta jumping: (a² + b²)/(ab + 1) is a square

#Vieta jumping #descent #Diophantine equations #extremal principle

Diophantine equations

Ex.2234Distance to $O_n$ and polar decomposition (statement)

#distance to O_n #singular values #polar decomposition

Topology of Mn(ℝ)

Ex.2235Every set of n integers contains a sum-free subset of more than n/3 elements

#probabilistic method #averaging #sum-free sets #modular arithmetic

Counting principles

Ex.2236Conway's soldiers cannot reach the fifth row

#weight function #potential #golden ratio #infinite board

Counting principles

Ex.2237The Erdős–Ko–Rado theorem

#set systems #double counting #intersecting families #Katona

Counting principles

Ex.2238A strictly increasing f with f(f(n)) = 3n

#functional equation #monotone #integers #ternary structure

Functional equations

Ex.2239The Erdős–Mordell inequality

#Erdős–Mordell #projections #pedal points #geometric inequality

Ex.2240Primitive roots exist modulo every prime

#primitive root #order #totient #polynomial roots

Ex.2241Continuous homomorphisms from GL(n,R) to R

#homomorphism #commutator subgroup #determinant #continuous characters

Applications and the case n = 2

Ex.2242No continuous square root on GL(2,R)

#square root #rotations #commutant #non-existence

Applications and the case n = 2

Ex.2243The Dirac belt trick: a 2Ï€ rotation is non-trivial, a 4Ï€ rotation is trivial

#homotopy #lifting #SO(3) #quaternions #Dirac belt

Exponential, logarithm and group topology

Ex.2244The square root is operator monotone

#Loewner order #operator monotone #square root #positive semidefinite

Rank, spectrum and norms

Ex.2245Solving a non-linear system via the continuity method

#continuity method #implicit function theorem #connectedness #nonlinear system

Topology of ℝⁿ