Olympiad problems

2001 Saint Petersburg Mathematical Olympiad, 9.2

Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers? [I]Proposed by F. Petrov[/i]

2019 Azerbaijan Junior NMO, 1

Tags: quadratic trinomial , quadratic , cell , trinomial

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.

2001 Saint Petersburg Mathematical Olympiad, 10.1

Tags: algebra , quadratic , quadratic trinomial , inequalities

Quadratic trinomials $f$ and $g$ with integer coefficients obtain only positive values and the inequality $\dfrac{f(x)}{g(x)}\geq \sqrt{2}$ is true $\forall x\in\mathbb{R}$. Prove that $\dfrac{f(x)}{g(x)}>\sqrt{2}$ is true $\forall x\in\mathbb{R}$ [I]Proposed by A. Khrabrov[/i]

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.)

2009 Bundeswettbewerb Mathematik, 2

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

1993 ITAMO, 2

Tags: quadratic trinomial , rational roots , rational , prime , quadratic , trinomial , number theory

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

2005 Junior Tuymaada Olympiad, 5

Tags: quadratic trinomial , trinomial , algebra

Given the quadratic trinomial $ f (x) = x ^ 2 + ax + b $ with integer coefficients, satisfying the inequality $ f (x) \geq - {9 \over 10} $ for any $ x $. Prove that $ f (x) \geq - {1 \over 4} $ for any $ x $.

2018 Irish Math Olympiad, 3

Tags: quadratic trinomial , trinomial , algebra

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

2015 India Regional MathematicaI Olympiad, 2

Tags: quadratic trinomial , integer , algebra , polynomial

Let $P(x)=x^{2}+ax+b$ be a quadratic polynomial where $a$ is real and $b \neq 2$, is rational. Suppose $P(0)^{2},P(1)^{2},P(2)^{2}$ are integers, prove that $a$ and $b$ are integers.

2022 Kyiv City MO Round 1, Problem 1

Tags: algebra , quadratic trinomial

Does there exist a quadratic trinomial $ax^2 + bx + c$ such that $a, b, c$ are odd integers, and $\frac{1}{2022}$ is one of its roots?

2018 Caucasus Mathematical Olympiad, 6

Tags: conic , algebra , quadratic trinomial , parabola

Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.

2019 Dutch IMO TST, 1

Tags: polynomial , trinomial , quadratic trinomial , algebra

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

2007 Junior Tuymaada Olympiad, 2

Tags: algebra , trinomial , quadratic trinomial

Two quadratic trinomials $ f (x) $ and $ g (x) $ differ from each other only by a permutation of coefficients. Could it be that $ f (x) \geq g (x) $ for all real $ x $?

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.

2022 Tuymaada Olympiad, 5

Tags: algebra , trinomial , quadratic trinomial

Prove that a quadratic trinomial $x^2 + ax + b (a, b \in R)$ cannot attain at ten consecutive integral points values equal to powers of $2$ with non-negative integral exponent. [i](F. Petrov )[/i]

2019 Dutch IMO TST, 1

Tags: algebra , polynomial , trinomial , quadratic trinomial

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

2022 Bulgarian Spring Math Competition, Problem 9.1

Tags: algebra , quadratic trinomial , quadratic , function

Let $f(x)$ be a quadratic function with integer coefficients. If we know that $f(0)$, $f(3)$ and $f(4)$ are all different and elements of the set $\{2, 20, 202, 2022\}$, determine all possible values of $f(1)$.

2002 Tuymaada Olympiad, 3

Tags: polynomial , algebra , integer polynomial , trinomial , quadratic trinomial , power of 2

Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?

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.