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

2012 Dutch BxMO/EGMO TST, 1
Tags: trinomial , quadratic , polynomial , root , algebra
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 $x = 7$?

2023 Canadian Mathematical Olympiad Qualification, 7
Tags: algebra , polynomial , trinomial , quadratic
(a) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 - 7x + 7 = 0$. Show that there exists a quadratic polynomial $f$ with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$. (b) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 -7x+4 = 0$. Show that there does not exist a quadratic polynomial $f $with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.

1976 Kurschak Competition, 3
Tags: polynomial , trinomial , algebra
Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$) then it can be written as the quotient of two polynomials whose coefficients are all positive.

2016 Saint Petersburg Mathematical Olympiad, 1
Tags: algebra , trinomial , quadratic trinomial , common real root
Given three quadratic trinomials $f, g, h$ without roots. Their elder coefficients are the same, and all their coefficients for x are different. Prove that there is a number $c$ such that the equations $f (x) + cg (x) = 0$ and $f (x) + ch (x) = 0$ have a common root.

2000 All-Russian Olympiad Regional Round, 9.1
Tags: algebra , trinomial
Misha solved the equation $x^2 + ax + b = 0$ and told Dima the set of four numbers - two roots and two coefficients of this equation (but not said which of them are roots and which are coefficients). Will he be able to Dima, find out what equation Misha solved if all the numbers in the set turned out to be different?

1985 Brazil National Olympiad, 4
Tags: diophantine equation , number theory , trinomial , integer
$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.

1955 Moscow Mathematical Olympiad, 307
Tags: trinomial , perfect square , number theory , algebra
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

1998 All-Russian Olympiad Regional Round, 9.5
Tags: algebra , trinomial
The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?

1995 All-Russian Olympiad Regional Round, 10.5
Tags: algebra , inequalities , trinomial
Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?