Olympiad problems

2010 Dutch IMO TST, 5

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

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 $w = f(u)$.

1952 Moscow Mathematical Olympiad, 221

Tags: algebra , root , trinomial

Prove that if for any positive $p$ all roots of the equation $ax^2 + bx + c + p = 0$ are real and positive then $a = 0$.

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 $x_1 \ge x_3 \ge x_2$.

1990 ITAMO, 5

Tags: number theory , divisible , trinomial

Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.

1949-56 Chisinau City MO, 17

Tags: algebra , trinomial , parameter

Prove that if the roots of the equation $x^2 + px + q = 0$ are real, then for any real number $a$ the roots of the equation $$x^2 + px + q + (x + a) (2x + p) = 0$$ are also real.

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 $x = 7$?

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.

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.

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 $b\ge 5$.

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , inequalities , trinomial

The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.

2009 Bundeswettbewerb Mathematik, 2

Tags: prime divisor , number theory , quadratic trinomial , trinomial , 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.

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 $| f '(0) | \le 8.$

2015 Puerto Rico Team Selection Test, 3

Tags: trinomial , quadratic , number theory , divisible

Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.

1990 All Soviet Union Mathematical Olympiad, 516

Tags: quadratic , root , rational , algebra , trinomial

Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.

2006 Junior Tuymaada Olympiad, 5

Tags: algebra , quadratic trinomial , trinomial , quadratic , constant , sides of a triangle

The quadratic trinomials $ f $, $ g $ and $ h $ are such that for every real $ x $ the numbers $ f (x) $, $ g (x) $ and $ h (x) $ are the lengths of the sides of some triangles, and the numbers $ f (x) -1 $, $ g (x) -1 $ and $ h (x) -1 $ are not the lengths of the sides of the triangle. Prove that at least of the polynomials $ f + g-h $, $ f + h-g $, $ g + h-f $ is constant.

2018 India PRMO, 9

Tags: trinomial , maximum , integer , root

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

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 $\frac{a+b+c}{b-a}$?

2018 Irish Math Olympiad, 3

Tags: algebra , trinomial , quadratic trinomial

Find all functions $f(x) = ax^2 + bx + c$, with $a \ne 0$, such that $f(f(1)) = f(f(0)) = f(f(-1))$ .

2017 Tuymaada Olympiad, 5

Tags: trinomial , integer polynomial , algebra , quadratic

Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

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?