Olympiad problems

2020 Jozsef Wildt International Math Competition, W27

Let $$P(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n$$ where $a_0,\ldots,a_n$ are integers. Show that if $P$ takes the value $2020$ for four distinct integral values of $x$, then $P$ cannot take the value $2001$ for any integral value of $x$. [i]Proposed by Ángel Plaza[/i]

2000 Saint Petersburg Mathematical Olympiad, 11.4

Tags: Polynomials , integer polynomials , number theory , Existence

Let $P(x)=x^{2000}-x^{1000}+1$. Prove that there don't exist 8002 distinct positive integers $a_1,\dots,a_{8002}$ such that $a_ia_ja_k|P(a_i)P(a_j)P(a_k)$ for all $i\neq j\neq k$. [I]Proposed by A. Baranov[/i]

2000 Saint Petersburg Mathematical Olympiad, 9.3

Tags: number theory , Polynomials , integer polynomials , Existence , St. Petersburg MO

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]

2023 Indonesia TST, A

Tags: algebra , polynomial , Polynomials , Integer Polynomial , integer polynomials , Indonesia , Indonesia TST

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 $a>b$

2023 Indonesia TST, A

Tags: algebra , polynomial , Polynomials , Integer Polynomial , integer polynomials , Indonesia , Indonesia TST

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 $a>b$

2014 Israel National Olympiad, 5

Tags: number theory , algebra , Polynomials , Integer Polynomial , integer polynomials , polynomial

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

2015 Indonesia MO Shortlist, A7

Tags: integer polynomials , algebra , polynomial

Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that \[ \sum_{i=1}^n P(i) | nP(n+1) \] for every $n \in \mathbb{N}$. Prove that there exists an integer $k \ge 0$ such that \[ P(n) = \binom{n+k}{n-1} P(1) \] for every $n \in \mathbb{N}$.