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

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 $4 b^2 -a^3$.

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 $$ (n - q)^2 - (m - p) (np - mq) = 0.$$

1918 Eotvos Mathematical Competition, 3
Tags: algebra , inequalities , trinomial
If $a, b,c,p,q, r $are real numbers such that, for every real number $x,$ $$ax^2 - 2bx + c \ge 0 \ \ and \ \ px^2 + 2qx + r \ge 0;$$ prove that then $$apx^2 + bqx + cr \ge 0$$ for all real $x$.

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 $f (m) = f (2015) \cdot f (2016)$.

1984 All Soviet Union Mathematical Olympiad, 383
Tags: integer root , trinomial , polynomial , game , combinatorics
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?

2005 Cuba MO, 2
Tags: algebra , trinomial , inequalities
Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.

1998 Denmark MO - Mohr Contest, 2
Tags: inequalities , trinomial , polynomial , algebra
For any real number$m$, the equation $$x^2+(m-2)x- (m+3)=0$$ has two solutions, denoted $x_1 $and $ x_2$. Determine $m$ such that $x_1^2+x_2^2$ is the minimum possible.

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 $b \ne d$.

VII Soros Olympiad 2000 - 01, 11.2
Tags: parameterization , algebra , trinomial
For all valid values ​​of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$

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?

1990 Greece Junior Math Olympiad, 4
Tags: algebra , trinomial
For which real values of $m$ does the equation $x^2-\frac{m^2+1}{m -1}x+2m+2=0$ has root $x=-1$?

2014 India PRMO, 6
Tags: trinomial , integer , root , minimum
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?

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 $x$.

2019 BMT Spring, 1
Tags: number theory , trinomial , algebra
How many integers $ x $ satisfy $ x^2 - 9x + 18 < 0 $?

1971 All Soviet Union Mathematical Olympiad, 149
Tags: algebra , polynomial , trinomial
Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $$(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$$ then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.

2010 Cuba MO, 1
Tags: algebra , trinomial , polynomial
Determine all the integers $a$ and $b$, such that $\sqrt{2010 + 2 \sqrt{2009}}$ be a solution of the equation $x^2 + ax + b = 0$. Prove that for such $a$ and $b$ the number$\sqrt{2010 - 2 \sqrt{2009}}$ is not a solution to the given equation.

2022 Junior Balkan Team Selection Tests - Moldova, 12
Tags: algebra , number theory , trinomial
Let $p$ and $q$ be two distinct integers. The square trinomial $x^2 + px + q$ is written on the board. At each step, a number is deleted: or the coefficient next to $x$, or the free term, and instead of the deleted number, a number is written, which is obtained from the deleted number by adding or subtracting the number $1$. After several such steps on the board, the square trinomial $x^2 + qx + p$ appeared. Show that at one stage a square trinomial was written on the board, both roots of which are integers.

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 $1$?

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 $x^2 + 3bx - 27 = 0$.

2019 Lusophon Mathematical Olympiad, 2
Tags: algebra , trinomial , rational roots
Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions: 1. $a + b + c = n$ 2. $ax^2 + bx + c = 0$ has rational roots.

2011 Flanders Math Olympiad, 1
Tags: algebra , inequalities , trinomial
Given are three numbers $a, b, c \in R-\{0\}$. The parabola with equation $y = ax^2+bx+c$ lies above the line $y = cx$. Prove that the parabola with equation $y = cx^2 - bx + a$ lies above the line $y = cx - b$.

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)

1998 Romania National Olympiad, 1
Tags: algebra , quadratic , trinomial
Find the integer numbers $a, b, c$ such that the function $f: R \to R$, $f(x) = ax^2 +bx + c$ satisfies the equalities : $$f(f(1) ))= f (f(2 ) )= f(f (3 ))$$

2015 Indonesia MO Shortlist, N2
Tags: integer , trinomial , area of a triangle , algebra , integer root , right triangle
Suppose that $a, b$ are natural numbers so that all the roots of $x^2 + ax - b$ and $x^2 - ax + b$ are integers. Show that exists a right triangle with integer sides, with $a$ the length of the hypotenuse and $b$ the area .

2005 Czech And Slovak Olympiad III A, 5
Tags: quadratic , quadratic trinomial , trinomial , algebra
Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$. Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$. (A double root is counted twice.)