Maths WorkshopMaths problems with full solutions

Equations and radicals

57 problems · Polynomial equations, radical equations and systems of algebraic equations.

Ex.180A nested radical
#radicals #identities
Equations and radicals
High school
High school
Ex.235A linear equation
#linear equation #solving #basic
Equations and radicals
High school
High school
Ex.236A quadratic equation
#quadratic equation #discriminant #roots
Equations and radicals
High school
High school
Ex.320An equation with absolute values
#absolute value #equations
Equations and radicals
High school
High school
Ex.462An exponential equation
#exponential equation #substitution #quadratic
Equations and radicals
High school
High school
Ex.463A logarithmic equation
#logarithmic equation #domain #exponential
Equations and radicals
High school
High school
Ex.464A quadratic with complex roots
#quadratic #complex roots #discriminant
Equations and radicals
High school
High school
Ex.465Solving by factoring
#factoring #roots #quadratic
Equations and radicals
High school
High school
Ex.466A simple radical equation
#radical equation #squaring #extraneous roots
Equations and radicals
High school
High school
Ex.467A system by substitution
#system #substitution #nonlinear
Equations and radicals
High school
High school
Ex.468A trigonometric equation
#trigonometric equation #solving #periodicity
Equations and radicals
High school
High school
Ex.632Cassini's identity and sums of Fibonacci squares
#Fibonacci numbers #matrices #determinant #induction
Equations and radicals
M1
M1
Ex.634Three telescoping sums
#telescoping #factorials #partial fractions
Equations and radicals
M1
M1
Ex.943A biquadratic equation
#biquadratic #substitution #real roots
Equations and radicals
High school
High school
Ex.944A cubic with three roots
#cubic #factoring #roots
Equations and radicals
L1
L1
Ex.945A cubic by factoring
#cubic #factoring #rational root
Equations and radicals
L1
L1
Ex.946Vieta's formulas
#Vieta #sum and product of roots #quadratic
Equations and radicals
L1
L1
Ex.947A quartic by factoring
#quartic #factoring #roots
Equations and radicals
L1
L1
Ex.948A cube root equation
#cube root #cubing #real solutions
Equations and radicals
L1
L1
Ex.949A radical inequality
#radical inequality #squaring #cases
Equations and radicals
L1
L1
Ex.950A radical equation with two square roots
#radical #two roots #conditions
Equations and radicals
L1
L1
Ex.951A radical equation by substitution
#radical #substitution #quadratic
Equations and radicals
L1
L1
Ex.952Two square roots
#radical equation #squaring twice #conditions
Equations and radicals
High school
High school
Ex.953A 3×3 linear system
#linear system #3 equations #elimination
Equations and radicals
High school
High school
Ex.954A nonlinear system
#nonlinear system #elimination #symmetry
Equations and radicals
L1
L1
Ex.955A symmetric system
#symmetric system #substitution #Vieta
Equations and radicals
L1
L1
Ex.1225The integer part of a telescoping sum
#recurrence #telescoping #floor #monotone sequence
Equations and radicals
Olympiad
Olympiad
Ex.1226cos(π/7) − cos(2π/7) + cos(3π/7) = 1/2
#trigonometry #telescoping #product-to-sum
Equations and radicals
Olympiad
Olympiad
Ex.1228A recurrence with a hidden invariant
#linear recurrence #invariant #characteristic equation
Equations and radicals
Olympiad
Olympiad
Ex.1230Hermite's identity for floor functions
#floor function #fractional part #Hermite identity
Equations and radicals
Olympiad
Olympiad
Ex.1232Two symmetric systems
#symmetric system #Vieta #sums of squares
Equations and radicals
M1
M1
Ex.1233Telescoping with arctangents and products
#telescoping #arctangent #products
Equations and radicals
Olympiad
Olympiad
Ex.1295Every integer divides some Fibonacci number
#Fibonacci numbers #periodicity #pigeonhole #modular arithmetic
Equations and radicals
Olympiad
Olympiad
Ex.1297How fast does a_{n+1} = a_n + 1/a_n grow?
#sequences #growth #squaring trick #estimates
Equations and radicals
Olympiad
Olympiad
Ex.1299tan 20° tan 40° tan 80° and tan²20° + tan²40° + tan²80°
#trigonometry #tangent #triple angle #Vieta
Equations and radicals
Olympiad
Olympiad
Ex.1300Approximating √2 by 1 + 1/(1 + x)
#continued fractions #sequences #Pell equation #approximation
Equations and radicals
Olympiad
Olympiad
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
Olympiad
Olympiad
Ex.1666A cubic equation with parameter
#cubic #parameter #roots
Equations and radicals
L1
L1
Ex.1667A cyclotomic equation
#roots of unity #cyclotomic #complex
Equations and radicals
Olympiad
Olympiad
Ex.1668A double radical
#double radical #squaring #verification
Equations and radicals
Olympiad
Olympiad
Ex.1669A functional equation
#functional equation #polynomial solutions #algebraic
Equations and radicals
L1
L1
Ex.1670An equation with the floor function
#floor function #integer part #olympiad
Equations and radicals
Olympiad
Olympiad
Ex.1671A radical equation with parameter
#parameter #radical equation #conditions
Equations and radicals
L1
L1
Ex.1672Polynomial divisibility and roots
#polynomial #divisibility #factor theorem
Equations and radicals
L1
L1
Ex.1673A polynomial equation with parameter
#parameter #roots #discriminant
Equations and radicals
L1
L1
Ex.1674Counting real roots
#polynomial #real roots #derivative
Equations and radicals
L1
L1
Ex.1675A quadratic with parameter
#parameter #quadratic #discriminant
Equations and radicals
L1
L1
Ex.1676A quartic with no real roots
#quartic #real roots #sum of squares
Equations and radicals
L1
L1
Ex.1677A radical equation leading to a quartic
#radical #quartic #substitution #extraneous roots
Equations and radicals
L1
L1
Ex.1678A system with an inequality
#system #inequality #integers
Equations and radicals
Olympiad
Olympiad
Ex.1679A system depending on a parameter
#parameter #linear system #discussion
Equations and radicals
L1
L1
Ex.1680A quadratic system
#nonlinear system #elimination #olympiad
Equations and radicals
Olympiad
Olympiad
Ex.1970gcd(F_m, F_n) = F_gcd(m,n)
#Fibonacci numbers #gcd #Euclidean algorithm #identities
Equations and radicals
Olympiad
Olympiad
Ex.2132A radical equation with parameter (advanced)
#parameter #radical equation #discussion
Equations and radicals
L1
L1
Ex.2133A nonlinear 3×3 system
#nonlinear system #olympiad #symmetry
Equations and radicals
Olympiad
Olympiad
Ex.2134A cyclic system
#cyclic system #symmetric #olympiad
Equations and radicals
Olympiad
Olympiad
Ex.2135A symmetric 3-variable system
#symmetric system #Vieta #olympiad
Equations and radicals
Olympiad
Olympiad