Olympiad problems

2000 Czech and Slovak Match, 4

Let $P(x)$ be a polynomial with integer coefficients. Prove that the polynomial $Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1$ has no integer roots.

1995 India Regional Mathematical Olympiad, 4

Tags: integer root , quadratic equation , eisenstein , algebra

Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.

2015 Indonesia MO Shortlist, N2

Tags: integer root , integer , area of a triangle , trinomial , right triangle , algebra

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 .

2015 Irish Math Olympiad, 9

Tags: integer root , polynomial , integer polynomial , algebra

Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.

2020 Iran MO (3rd Round), 2

Tags: polynomial , integer root , number theory

Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)

1998 Czech and Slovak Match, 2

Tags: integer root , integer polynomial , algebra

A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.

2018 Pan-African Shortlist, C2

Tags: game , polynomial , integer root , algebra , combinatorics

Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree $n$. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if $n$ is odd; otherwise Adamu plays first. Prove that: [list] [*] Adamu has a winning strategy if $n$ is odd. [*] Afaafa has a winning strategy if $n$ is even. [/list]

1941 Moscow Mathematical Olympiad, 077

Tags: integer root , integer polynomial , polynomial , algebra

A polynomial $P(x)$ with integer coefficients takes odd values at $x = 0$ and $x = 1$. Prove that $P(x)$ has no integer roots.

2013 IFYM, Sozopol, 6

Tags: equation , integer root , algebra

For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots?

1991 Nordic, 4

Tags: number theory , integer polynomial , integer root

Let $f(x)$ be a polynomial with integer coefficients. We assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, ... , n+k -1$ so that none of the numbers $f(n), f(n+ 1),... , f(n + k - 1)$ is divisible by $k$. Show that the zeroes of $f(x)$ are not integers.

2021 India National Olympiad, 2

Tags: algebra , cubic , integer root , vieta

Find all pairs of integers $(a,b)$ so that each of the two cubic polynomials $$x^3+ax+b \, \, \text{and} \, \, x^3+bx+a$$ has all the roots to be integers. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

1990 All Soviet Union Mathematical Olympiad, 533

Tags: polynomial , game , algebra , game strategy , integer root , cubic

A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?

2006 Spain Mathematical Olympiad, 1

Tags: algebra , integer root , integer polynomial , polynomial

Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$ does not have integer roots.

1989 Nordic, 1

Tags: integer polynomial , integer root , algebra

Find a polynomial $P$ of lowest possible degree such that (a) $P$ has integer coefficients, (b) all roots of $P$ are integers, (c) $P(0) = -1$, (d) $P(3) = 128$.

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.

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 Ecuador NMO (OMEC), 4

Tags: polynomial , algebra , integer root

Let $k$ be a real number. Show that the polynomial $p (x) = x^3-24x + k$ has at most an integer root.