Olympiad problems

2023 SG Originals, Q6

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

2024 Korea National Olympiad, 4

Find the smallest positive integer $ k \geq 2 $ for which there exists a polynomial $ f(x) $ of degree $ k $ with integer coefficients and a leading coefficient of $ 1 $ that satisfies the following condition: (Condition) For any two integers $ m $ and $ n $, if $ f(m) - f(n) $ is a multiple of $ 31 $, then $ m - n $ is a multiple of $ 31 $.

number theory modulo polynomial with integer coeffi invariants modular arithmetic polynomials

2021 Azerbaijan Senior NMO, 5

Define $P(x)=((x-a_1)(x-a_2)...(x-a_n))^2 +1$, where $a_1,a_2...,a_n\in\mathbb{Z}$ and $n\in\mathbb{N^+}$. Prove that $P(x)$ couldn't be expressed as product of two non-constant polynomials with integer coefficients.

integer coefficients polynomial polynomial with integer coeffi algebra polynomials

2023 IRN-SGP-TWN Friendly Math Competition, 6

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

functional equation polynomial number theory polynomial with integer coeffi algebra functional equations modular arithmetic polynomials