Olympiad problems

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Found problems: 3

2014 India IMO Training Camp, 1

Tags: polynomial , number theory , FLT

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

Tags: USEMO , number theory , Fermat s Little Theorem , Hi , FLT

Prove that for any prime $p,$ there exists a positive integer $n$ such that \[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\] [i]Robin Son[/i]

Tags: polynomial , number theory , FLT

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.