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

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.

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 $[\beta, \beta + \alpha]$.

2011 Brazil Team Selection Test, 1
Tags: trinomial , polynomial , algebra
Let $P_1$, $P_2$ and $P_3$ be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial $P_1 + P_2 + P_3$ has also real roots.

2003 Cuba MO, 1
Tags: algebra , polynomial , trinomial
The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

1954 Moscow Mathematical Olympiad, 285
Tags: algebra , quadratic , trinomial , absolute value , root
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 $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

2014 India PRMO, 9
Tags: algebra , trinomial , root
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$?

2006 Estonia Team Selection Test, 1
Tags: integer , algebra , quadratic , trinomial , integer polynomial
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$. a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$. b) Find the sum of the other components of all such pairs of numbers.

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} $

2009 Puerto Rico Team Selection Test, 4
Tags: algebra , trinomial , quadratic polynomial
Find all integers $ b$ and $ c$ such that the equation $ x^2 - bx + c = 0$ has two real roots $ x_1, x_2$ satisfying $ x_1^2 + x_2^2 = 5$.

2001 Abels Math Contest (Norwegian MO), 1a
Tags: trinomial , quadratic polynomial , algebra
Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.

2005 Junior Balkan Team Selection Tests - Moldova, 8
Tags: inequalities , trinomial , quadratic , quadratic polynomial , function , algebra
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter. Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.

1996 All-Russian Olympiad Regional Round, 9.1
Tags: algebra , trinomial
Find all pairs of square trinomials $x^2 + ax + b$, $ x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.

2013 Cuba MO, 1
Tags: algebra , trinomial
Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.

2018 Istmo Centroamericano MO, 4
Tags: trinomial , quadratic , perfect square , number theory
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.

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.

2001 All-Russian Olympiad Regional Round, 10.5
Tags: algebra , polynomial , number theory , trinomial
Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.

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?

1993 ITAMO, 2
Tags: quadratic , quadratic trinomial , trinomial , number theory , rational roots , rational , prime
Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

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?

1949-56 Chisinau City MO, 18
Tags: algebra , trinomial
Prove that if the numbers $a, b, c$ are the lengths of the sides of some nondegenerate triangle, then the equation $$b^2x^2 + (b^2 + c^2 - a^2) x + c^2 = 0$$ has imaginary roots.

1999 Estonia National Olympiad, 2
Tags: algebra , parameter , trinomial
Find all values of $a$ such that absolute value of one of the roots of the equation $x^2 + (a - 2)x - 2a^2 + 5a - 3 = 0$ is twice of absolute value of the other root.

1953 Poland - Second Round, 1
Tags: algebra , trinomial , polynomial
Prove that the equation $$ (x - a) (x - c) + 2 (x - b) (x - d) = 0,$$ in which $ a < b < c < d $, has two real roots.

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.