This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 111

1968 IMO Shortlist, 6
Tags: algebra , polynomial , equation , roots , Real Roots , IMO Shortlist
If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation \[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\] has at least $n - 1$ real roots.

2017 India PRMO, 4
Tags: trinomial , roots , Integers , algebra
Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$ ?

2005 Greece Team Selection Test, 1
Tags: algebra , polynomial , roots
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

2017 Vietnamese Southern Summer School contest, Problem 2
Tags: algebra , polynomial , roots , equation
Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

2021 German National Olympiad, 1
Tags: quadratics , algebra , algebra proposed , quadratic equation , roots
Determine all real numbers $a,b,c$ and $d$ with the following property: The numbers $a$ and $b$ are distinct roots of $2x^2-3cx+8d$ and the numbers $c$ and $d$ are distinct roots of $2x^2-3ax+8b$.

2005 Greece Team Selection Test, 1
Tags: algebra , polynomial , roots
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

1989 IMO Longlists, 8
Tags: algebra , polynomial , roots , equation , IMO Shortlist , Diophantine equation
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2019 Middle European Mathematical Olympiad, 2
Tags: algebra , polynomial , memo , MEMO 2019 , roots
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$. [i]Proposed by Walther Janous, Austria[/i]

2013 South East Mathematical Olympiad, 1
Tags: inequalities , algebra , cubic function , roots
Let $a,b$ be real numbers such that the equation $x^3-ax^2+bx-a=0$ has three positive real roots . Find the minimum of $\frac{2a^3-3ab+3a}{b+1}$.

1953 Moscow Mathematical Olympiad, 253
Tags: inequalities , roots , trinomial , algebra
Given the equations (1) $ax^2 + bx + c = 0$ (2)$ -ax^2 + bx + c = 0$ prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.

2016 Junior Regional Olympiad - FBH, 2
Tags: roots , algebra
If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$

2011 VTRMC, Problem 7
Tags: algebra , polynomial , Polynomials , roots
Let $P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1$ be a polynomial where the $a_i~(1\le i\le97)$ are real numbers. Prove that the equation $P(x)=0$ has at least one nonreal root.

1961 Putnam, B6
Tags: Putnam , function , differential equation , roots
Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.

1968 Vietnam National Olympiad, 1
Tags: function , roots , analysis , algebra , inequalities
Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$. i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$. ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$. Show that $f(x)$ has two roots that are in between $-1$ and $1$.

2015 Caucasus Mathematical Olympiad, 2
Tags: algebra , roots , trinomial
Let $a$ and $b$ be arbitrary distinct numbers. Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.

2006 IMO, 5
Tags: algebra , polynomial , roots , IMO , IMO 2006 , IMO Shortlist , Dan Schwarz
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

1969 IMO Longlists, 14
Tags: quadratics , algebra , Inequality , roots , polynomial , IMO Longlist , IMO Shortlist
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

1966 IMO Longlists, 48
Tags: algebra , equation , number theory , roots , parametric equation , IMO Longlist , IMO Shortlist
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2025 VJIMC, 4
Tags: complex analysis , holomorphic , VJIMC , Taylor series , roots , function
Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.

1976 IMO Longlists, 11
Tags: algebra , polynomial , recurrence relation , roots , IMO , chebyshev polynomial , imo 1976
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

2018 Middle European Mathematical Olympiad, 2
Tags: algebra , polynomial , roots
Let $P(x)$ be a polynomial of degree $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients .

1954 Moscow Mathematical Olympiad, 285
Tags: algebra , quadratics , trinomial , absolute value , roots
The absolute values of all roots of the quadratic equation $x^2+Ax+B = 0$ and $x^2+Cx+D = 0$ are less then $1$. Prove that so are absolute values of the roots of the quadratic equation $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

Kvant 2023, M2738
Tags: algebra , roots
The real numbers $a_1,a_2,a_3$ and $b{}$ are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots, $c_1,c_2,c_3.$ Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\][i]Proposed by A. Antropov and K. Sukhov[/i]

1973 IMO Shortlist, 11
Tags: algebra , polynomial , roots , minimum value , IMO , IMO 1973 , coefficients
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

2018 Junior Regional Olympiad - FBH, 3
Tags: Compare , roots
Let $a$, $b$ and $m$ be three positive real numbers and $a>b$. Which of the numbers $A=\sqrt{a+m}-\sqrt{a}$ and $B=\sqrt{b+m}-\sqrt{b}$ is bigger: