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: 109

2002 Tuymaada Olympiad, 3
Tags: algebra , polynomial , integer polynomial , trinomial , quadratic trinomial , power of 2
Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?

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.

2005 Junior Tuymaada Olympiad, 5
Tags: quadratic trinomial , trinomial , algebra
Given the quadratic trinomial $ f (x) = x ^ 2 + ax + b $ with integer coefficients, satisfying the inequality $ f (x) \geq - {9 \over 10} $ for any $ x $. Prove that $ f (x) \geq - {1 \over 4} $ for any $ x $.

2005 Estonia National Olympiad, 4
Tags: equation , algebra , trinomial
Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

1990 Bundeswettbewerb Mathematik, 1
Tags: trinomial , quadratic polynomial , integer polynomial , algebra
Consider the trinomial $f(x) = x^2 + 2bx + c$ with integer coefficients $b$ and $c$. Prove that if $f(n) \ge 0$ for all integers $n$, then $f(x) \ge 0$ even for all rational numbers $x$.

2006 Junior Tuymaada Olympiad, 5
Tags: algebra , quadratic trinomial , trinomial , quadratic , constant , sides of a triangle
The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

2016 Hanoi Open Mathematics Competitions, 14
Tags: algebra , polynomial , trinomial
Let $f (x) = x^2 + px + q$, where $p, q$ are integers. Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$.

1985 All Soviet Union Mathematical Olympiad, 400
Tags: trinomial , integer , maximum , polynomial , algebra
The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

2017 Tuymaada Olympiad, 5
Tags: trinomial , integer polynomial , algebra , quadratic
Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

2008 Cuba MO, 1
Tags: algebra , inequalities , trinomial
Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$

2006 All-Russian Olympiad Regional Round, 11.2
Tags: algebra , polynomial , trinomial
Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.

VMEO III 2006, 10.4
Tags: algebra , trinomial
Find the least real number $\alpha$ such that there is a real number $\beta$ so that for all triples of real numbers $(a, b,c)$ satisfying $2006a + 10b + c = 0$, the equation $ax^2 + bx + c = 0$ always has real root in the interval $[\beta, \beta + \alpha]$.

2013 Czech-Polish-Slovak Match, 1
Tags: perfect square , trinomial , number theory , integer polynomial
Let $a$ and $b$ be integers, where $b$ is not a perfect square. Prove that $x^2 + ax + b$ may be the square of an integer only for finite number of integer values of $x$. (Martin Panák)

2019 Dutch IMO TST, 1
Tags: trinomial , quadratic trinomial , algebra , polynomial
Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

1999 Estonia National Olympiad, 2
Tags: trinomial , quadratic polynomial , algebra
It is known that the quadratic equations $x^2 + 6x + 4a = 0$ and $x^2 + 2bx - 12 = 0$ have a common solution. Prove that then there is a common solution to the quadratic equations $x^2 + 9x + 9a = 0$ and $x^2 + 3bx - 27 = 0$.

1994 All-Russian Olympiad Regional Round, 9.3
Tags: number theory , digit , trinomial
Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

2014 Hanoi Open Mathematics Competitions, 9
Tags: trinomial , algebra , quadratic
Determine all real numbers $a, b,c$ such that the polynomial $f(x) = ax^2 + bx + c$ satisfi es simultaneously the folloving conditions $\begin{cases} |f(x)| \le 1 \text{ for } |x | \le 1 \\ f(x) \ge 7 \text{ for } x \ge 2 \end{cases} $

2006 All-Russian Olympiad Regional Round, 10.4
Tags: algebra , polynomial , trinomial
Given $n > 1$ monic square trinomials $x^2 - a_1x + b_1$,$...$, $x^2-a_nx + b_n$, and all $2n$ numbers are $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n$ are different. Can it happen that each of the numbers $a_1$,$...$, $a_n$, $b_1$,$...$, $b_n is the root of one of these trinomials?

2018 Istmo Centroamericano MO, 4
Tags: trinomial , quadratic , number theory , perfect square
Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

2003 All-Russian Olympiad Regional Round, 11.5
Tags: algebra , trinomial
Square trinomials $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ are such that the equation $P(Q(x)) = Q(P(x))$ has no real roots. Prove that $b \ne d$.

1984 All Soviet Union Mathematical Olympiad, 383
Tags: trinomial , polynomial , game , combinatorics , integer root
The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?

2010 Dutch IMO TST, 5
Tags: number theory , trinomial
The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

2006 Thailand Mathematical Olympiad, 15
Tags: trinomial , number theory
How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer rooot?

2013 Poland - Second Round, 1
Tags: trinomial , divisibility , number theory
Let $b$, $c$ be integers and $f(x) = x^2 + bx + c$ be a trinomial. Prove, that if for integers $k_1$, $k_2$ and $k_3$ values of $f(k_1)$, $f(k_2)$ and $f(k_3)$ are divisible by integer $n \neq 0$, then product $(k_1 - k_2)(k_2 - k_3)(k_3 - k_1)$ is divisible by $n$ too.

1951 Poland - Second Round, 4
Tags: algebra , trinomial
Prove that if equations $$x^2 + mx + n = 0 \,\,\,\, and\,\, \,\, x^2 + px + q = 0$$ have a common root, there is a relationship between the coefficients of these equations $$ (n - q)^2 - (m - p) (np - mq) = 0.$$