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: 109

1990 Bundeswettbewerb Mathematik, 1
Tags: algebra , trinomial , quadratic polynomial , integer polynomial
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 $x$.

2010 All-Russian Olympiad Regional Round, 11.7
Tags: algebra , trinomial
Integers $a,b,c$ are such that the values of the trinomials $bx^2+cx+a$ and $cx^2+ax+b$ at $x=1234$ coincide. Can the first trinomial at $x = 1$ take the value $2009$?

1986 All Soviet Union Mathematical Olympiad, 418
Tags: algebra , polynomial , composite , trinomial
The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.

2016 Costa Rica - Final Round, F1
Tags: inequalities , trinomial , algebra
Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.

2016 Saint Petersburg Mathematical Olympiad, 1
Tags: trinomial , quadratic trinomial , common real root , algebra
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.

2007 Junior Balkan Team Selection Tests - Moldova, 6
Tags: angle , quadratic , trinomial
The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$

2008 Cuba MO, 1
Tags: inequalities , algebra , 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 $S(p) \ge ra^2$

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

1985 All Soviet Union Mathematical Olympiad, 400
Tags: algebra , polynomial , trinomial , integer , maximum
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 $|P(x)|<50$.