Olympiad problems

1947 Putnam, B5

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

2016 PAMO, 3

Tags: integer polynomial , greatest common divisor , number theory

For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.

2023 Indonesia TST, A

Tags: polynomial , integer polynomial , algebra

Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation: \[Q(a+b) = \frac{P(a) - P(b)}{a - b}\] $\forall a, b \in \mathbb{Z}^+$ and $a>b$

2015 Saudi Arabia Pre-TST, 3.2

Tags: integer polynomial , algebra , polynomial

Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients. (Le Anh Vinh)

2019 ISI Entrance Examination, 7

Tags: integer polynomial , polynomial , algebra

Let $f$ be a polynomial with integer coefficients. Define $$a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,$$ and $~a_n = f(a_{n-1})$ for $n \geqslant 3$. If there exists a natural number $k \geqslant 3$ such that $a_k = 0$, then prove that either $a_1=0$ or $a_2=0$.

2018 Iran Team Selection Test, 3

Tags: integer polynomial , number theory , polynomial

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $ [i]Proposed by Mojtaba Zare[/i]

2020 USA IMO Team Selection Test, 5

Tags: polynomial , integer polynomial , inequalities , number theory

Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

2000 Czech and Slovak Match, 4

Tags: integer polynomial , integer root , algebra , polynomial

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.

2021 Taiwan Mathematics Olympiad, 5.

Tags: integer polynomial , polynomial , algebra

Let $n$ be a given positive integer. Alice and Bob play a game. In the beginning, Alice determines an integer polynomial $P(x)$ with degree no more than $n$. Bob doesn’t know $P(x)$, and his goal is to determine whether there exists an integer $k$ such that no integer roots of $P(x) = k$ exist. In each round, Bob can choose a constant $c$. Alice will tell Bob an integer $k$, representing the number of integer $t$ such that $P(t) = c$. Bob needs to pay one dollar for each round. Find the minimum cost such that Bob can guarantee to reach his goal. [i]Proposed by ltf0501[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 11.6

Tags: integer polynomial , polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients. It is known that the number $\sqrt2+\sqrt3$ is its root. Prove that the number $\sqrt2-\sqrt3$ is also its root.

2021 Estonia Team Selection Test, 2

Tags: integer polynomial , algebra , polynomial

Find all polynomials $P(x)$ with integral coefficients whose values at points $x = 1, 2, . . . , 2021$ are numbers $1, 2, . . . , 2021$ in some order.

2015 Latvia Baltic Way TST, 3

Tags: algebra , integer polynomial , polynomial

Prove that there does not exist a polynomial $P (x)$ with integer coefficients and a natural number $m$ such that $$x^m + x + 2 = P(P(x))$$ holds for all integers $x$.

2013 Balkan MO Shortlist, A5

Tags: algebra , divide , integer polynomial , polynomial

Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.

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.

2014 Israel National Olympiad, 5

Tags: integer polynomial , algebra , polynomial , number theory

Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.

2011 Dutch IMO TST, 4

Tags: algebra , integer polynomial , polynomial

Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.

2021 239 Open Mathematical Olympiad, 1

Tags: prime , integer , integer polynomial , polynomial , algebra

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

1988 Austrian-Polish Competition, 1

Tags: integer polynomial , algebra , polynomial , integer

Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.

1988 Tournament Of Towns, (180) 3

Tags: polynomial , integer polynomial , algebra

It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .

2000 Saint Petersburg Mathematical Olympiad, 9.3

Tags: number theory , polynomial , existence , integer polynomial

Let $P(x)=x^{2000}-x^{1000}+1$. Do there exist distinct positive integers $a_1,\dots,a_{2001}$ such that $a_ia_j|P(a_i)P(a_j)$ for all $i\neq j$? [I]Proposed by A. Baranov[/i]

1986 All Soviet Union Mathematical Olympiad, 439

Tags: integer polynomial , algebra , polynomial

Let us call a polynomial [i]admissible[/i] if all it's coefficients are $0, 1, 2$ or $3$. For given $n$ find the number of all the [i]admissible [/i] polynomials $P$ such, that $P(2) = n$.

2016 Thailand TSTST, 1

Tags: integer polynomial , polynomial , number theory , algebra

Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.

2019 USA TSTST, 6

Tags: integer polynomial , fibonacci , algebra

Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant? (A [i]Fibonacci number[/i] is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.) [i]Nikolai Beluhov[/i]

2022 Turkey Team Selection Test, 6

Tags: polynomial , integer polynomial , algebra , number theory

For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?

2011 Dutch IMO TST, 4

Tags: integer polynomial , polynomial , algebra

Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.