Maths WorkshopMaths problems with full solutions

Algebra problems

304 problems · Equations, inequalities, polynomials, complex numbers and algebraic structures.

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Ex.179A classic inequality

#inequalities #AM-GM

Ex.180A nested radical

#radicals #identities

Equations and radicals

Ex.190A power of 1 + i

#complex numbers

Complex numbers

Ex.191Squares and pairwise products

#inequalities #squares

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.235A linear equation

#linear equation #solving #basic

Equations and radicals

Ex.236A quadratic equation

#quadratic equation #discriminant #roots

Equations and radicals

Ex.241Additive and multiplicative notations

#notation #abelian group #convention

Ex.242Checking the group axioms

#group axioms #examples #counterexamples

Ex.261Degree of sums and products

#degree #operations #polynomial

Ex.276Checking the ring axioms

#ring axioms #examples #counterexamples

Rings and fields

Ex.320An equation with absolute values

#absolute value #equations

Equations and radicals

Ex.333Remainder of a polynomial

#polynomials #euclidean division

Ex.336Sum of cubes of the roots

#vieta #polynomials

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.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.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.489The AM-GM inequality

#AM-GM #inequality #arithmetic and geometric means

Ex.490The triangle inequality

#triangle inequality #absolute value #real numbers

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

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

#sum of squares #SOS #real numbers

Ex.699AM-GM for three numbers

#inequalities #AM-GM

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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.2225Chebyshev polynomials and rational values of cosine

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

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