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

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Found problems: 117

2019 Middle European Mathematical Olympiad, 2
Tags: algebra , root , polynomial
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]

2020 German National Olympiad, 3
Tags: inequalities , root , algebra , polynomial
Show that the equation \[x(x+1)(x+2)\dots (x+2020)-1=0\] has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies \[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]

1976 IMO, 2
Tags: recurrence relation , polynomial , root , chebyshev polynomial , algebra
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.

2011 VTRMC, Problem 7
Tags: algebra , polynomial , root
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.

1949-56 Chisinau City MO, 10
Tags: root , rational , algebra
Get rid of irrationality in the denominator of a fraction $$\frac{1}{\sqrt[3]{4}+\sqrt[3]{2}+2}$$.

1985 Spain Mathematical Olympiad, 7
Tags: algebra , polynomial , root , integer polynomial , polynomial equation
Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.

Kvant 2023, M2738
Tags: algebra , root
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]

1952 Moscow Mathematical Olympiad, 221
Tags: root , trinomial , algebra
Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.

2005 Greece Team Selection Test, 1
Tags: polynomial , root , algebra
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.

2000 Estonia National Olympiad, 3
Tags: algebra , polynomial , root , integer
Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.

2018 Junior Regional Olympiad - FBH, 3
Tags: compare , root
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:

1988 IMO Shortlist, 16
Tags: algebra , vieta , inequality , root , polynomial
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

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

1971 IMO Longlists, 31
Tags: polynomial , equation , algebra , root
Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

2020/2021 Tournament of Towns, P4
Tags: algebra , root
It is well-known that a quadratic equation has no more than 2 roots. Is it possible for the equation $\lfloor x^2\rfloor+px+q=0$ with $p\neq 0$ to have more than 100 roots? [i]Alexey Tolpygo[/i]

1981 Putnam, A5
Tags: polynomial , root
Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

2012 Dutch BxMO/EGMO TST, 1
Tags: algebra , polynomial , trinomial , quadratic , root
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

1940 Putnam, B5
Tags: polynomial , rational , root , algebra
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

1996 German National Olympiad, 6a
Tags: polynomial , root , algebra
Prove the following statement: If a polynomial $p(x) = x^3 + Ax^2 + Bx +C$ has three real positve roots at least two of which are distinct, then $A^2 +B^2 +18C > 0$.

2015 Germany Team Selection Test, 1
Tags: polynomial , algebra , root , real number , integer
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

2014 India PRMO, 17
Tags: minimum , algebra , integer , root
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?

2016 India PRMO, 9
Tags: algebra , trinomial , root
Let $a$ and $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4 b^2 -a^3$.

2010 Hanoi Open Mathematics Competitions, 7
Tags: number theory , prime , quadratic , root
Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.

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

1999 Bosnia and Herzegovina Team Selection Test, 1
Tags: algebra , root , equation , real root
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions