Olympiad problems

2006 Thailand Mathematical Olympiad, 15

How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer rooot?

1979 Swedish Mathematical Competition, 5

Tags: algebra , trinomial

Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.

2013 Czech-Polish-Slovak Match, 1

Tags: perfect square , trinomial , integer polynomial , number theory

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)

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 Junior Balkan Team Selection Tests - Moldova, 12

Tags: algebra , trinomial , number theory

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.

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 $c> 0$.

2013 Tournament of Towns, 5

Tags: algebra , integer root , trinomial

A quadratic trinomial with integer coefficients is called [i]admissible [/i] if its leading coefficient is $1$, its roots are integers and the absolute values of coefficients do not exceed $2013$. Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.

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?

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.

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 $4b^2 -a^3$.

1999 Estonia National Olympiad, 2

Tags: algebra , trinomial , quadratic polynomial

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

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , root , trinomial

Let $a$ and $b$ be arbitrary distinct numbers. Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.

1990 ITAMO, 5

Tags: number theory , divisible , trinomial

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

1976 Kurschak Competition, 3

Tags: algebra , polynomial , trinomial

Prove that if the quadratic $x^2 +ax+b$ is always positive (for all real $x$) then it can be written as the quotient of two polynomials whose coefficients are all positive.

2010 Cuba MO, 1

Tags: algebra , polynomial , trinomial

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.

VII Soros Olympiad 2000 - 01, 11.2

Tags: algebra , parameterization , trinomial

For all valid values of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$

1953 Moscow Mathematical Olympiad, 253

Tags: inequalities , algebra , root , trinomial

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 $x_1 \ge x_3 \ge x_2$.

1997 Estonia National Olympiad, 2

Tags: trinomial , quadratic polynomial , algebra

Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.

2015 Puerto Rico Team Selection Test, 3

Tags: number theory , divisible , trinomial , quadratic

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 $5$.

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?

2014 India PRMO, 6

Tags: minimum , trinomial , integer , root

What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer 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 $x^2_2$.

1984 All Soviet Union Mathematical Olympiad, 383

Tags: polynomial , trinomial , game , integer root , 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?

2018 India PRMO, 9

Tags: root , integer , trinomial , maximum

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