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

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

1902 Eotvos Mathematical Competition, 1
Tags: algebra , trinomial
Prove that any quadratic expression $$Q(x) = Ax^2 + Bx + C$$ (a) can be put into the form $$Q(x) = k \frac{x(x- 1)}{1 \cdot 2} + \ell x + m$$ where $k, \ell, m$ depend on the coefficients $A,B,C$ and (b) $Q(x)$ takes on integral values for every integer $x$ if and only if $k, \ell, m$ are integers.

2015 Indonesia MO Shortlist, N2
Tags: trinomial , area of a triangle , algebra , integer root , right triangle , integer
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 .

2003 Switzerland Team Selection Test, 7
Tags: trinomial , polynomial , quadratic polynomial , algebra
Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.

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?

1953 Moscow Mathematical Olympiad, 253
Tags: inequalities , root , trinomial , algebra
Given the equations (1) $ax^2 + bx + c = 0$ (2)$ -ax^2 + bx + c = 0$ prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.

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 $f(x)f(x -1) < 0$.

2019 Azerbaijan Junior NMO, 1
Tags: quadratic , trinomial , quadratic trinomial , cell
A $6\times6$ square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are $108$ coefficents in total, and these coefficents are chosen from the set $[-66;47]$, and each coefficient is different from each other. Prove that there exists at least one column such that the polynomial you get by summing the six trinomials in that column has a real root.

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 $x = 7$?

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 $c> 0$.

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 $4 b^2 -a^3$.

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 $A(m) = A(m + 1) = 0$.

1996 Chile National Olympiad, 4
Tags: algebra , quadratic , trinomial
Let $a, b, c$ be naturals. The equation $ax^2-bx + c = 0$ has two roots at $[0, 1]$. Prove that $a\ge 5$ and $b\ge 5$.

1990 All Soviet Union Mathematical Olympiad, 529
Tags: inequalities , trinomial , algebra
A quadratic polynomial $p(x)$ has positive real coefficients with sum $1$. Show that given any positive real numbers with product $1$, the product of their values under $p$ is at least $1$.

1976 Dutch Mathematical Olympiad, 4
Tags: algebra , parameter , parameterization , equation , trinomial
For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?

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 $x=-1$?

2009 Bundeswettbewerb Mathematik, 2
Tags: number theory , quadratic trinomial , trinomial , prime , divisor , prime divisor
Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

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 $x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0$.

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 $c$.

1990 ITAMO, 5
Tags: number theory , divisible , trinomial
Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.

1979 Chisinau City MO, 180
Tags: trinomial , algebra , derivative
It is known that for $0\le x \le 1$ the square trinomial $f (x)$ satisfies the condition $|f(x) | \le 1$. Show that $| f '(0) | \le 8.$

2007 Junior Balkan Team Selection Tests - Moldova, 6
Tags: trinomial , angle , quadratic
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 $18^o$

2019 Paraguay Mathematical Olympiad, 1
Tags: algebra , trinomial , quadratic
Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem $-1$ and $-3$. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem $16$ and $-2$. What are the correct results of the problem?

1979 Chisinau City MO, 171
Tags: trinomial , algebra
Are there numbers $a, b$ such that $| a -b |\le 1979$ and the equation $ax^2 + (a + b) x + b = x$ has no roots?

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.