Olympiad problems

1964 Swedish Mathematical Competition, 3

Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.

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 $P(P(x))$.

2014 Swedish Mathematical Competition, 1

Tags: polynomial , algebra , integer polynomial

Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.

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.

1998 Singapore Team Selection Test, 3

Tags: integer polynomial , algebra , polynomial

Suppose $f(x)$ is a polynomial with integer coefficients satisfying the condition $0 \le f(c) \le 1997$ for each $c \in \{0, 1, ..., 1998\}$. Is is true that $f(0) = f(1) = ... = f(1998)$? (variation of [url=https://artofproblemsolving.com/community/c6h49788p315649]1997 IMO Shortlist p12[/url])

2021 Mediterranean Mathematics Olympiad, 1

Tags: algebra , integer polynomial , polynomial

Determine the smallest positive integer $M$ with the following property: For every choice of integers $a,b,c$, there exists a polynomial $P(x)$ with integer coefficients so that $P(1)=aM$ and $P(2)=bM$ and $P(4)=cM$. [i]Proposed by Gerhard Woeginger, Austria[/i]

1939 Moscow Mathematical Olympiad, 049

Tags: polynomial , odd , integer polynomial , even , algebra

Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.

2017 South Africa National Olympiad, 6

Tags: number theory , algebra , integer polynomial , polynomial

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.

1995 Poland - Second Round, 1

Tags: integer polynomial , polynomial , divisible , algebra

For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.

1999 Israel Grosman Mathematical Olympiad, 4

Tags: polynomial , integer polynomial , factoring , algebra

Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients. Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients

1992 Nordic, 2

Tags: integer polynomial , polynomial division , algebra

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.

2020 USA EGMO Team Selection Test, 6

Tags: integer polynomial , periodic sequence , modular arithmetic , polynomial , algebra

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2004 BAMO, 5

Tags: integer polynomial , irrational , algebra , polynomial

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

2010 Swedish Mathematical Competition, 3

Tags: polynomial , integer polynomial , algebra

Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .

1997 Austrian-Polish Competition, 5

Tags: polynomial , prime , integer polynomial , cubic

Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

2021 Saudi Arabia Training Tests, 35

Tags: polynomial , integer polynomial , algebra

Let $P (x)$ be a non constant integer polynomial and positive integer $n$. The sequence $a_0, a_1, ...$ is defined by $a_0 = n$ and $a_k = P (a_{k-1})$ for $k \ge 1$. Given that for each positive integer $b$, the sequence contains a $b$-th power of some positive integer greater than $1$. Prove that deg $P = 1$

2011 Ukraine Team Selection Test, 11

Tags: integer polynomial , algebra , inequalities , polynomial

Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds: $$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$

2022 Austrian MO National Competition, 4

Tags: polynomial , integer polynomial , number theory

Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$. [i](Walther Janous)[/i]

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.

2022 Federal Competition For Advanced Students, P2, 4

Tags: integer polynomial , number theory , polynomial