Olympiad problems

2019 USEMO, 4

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]

2023 Indonesia TST, N

Tags: number theory , Fermat s Little Theorem , primitive root , Primitive Roots

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

2024 Baltic Way, 17

Tags: number theory proposed , prime numbers , Fermat s Little Theorem , Divisibility , Bound

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

2005 IMO, 4

Tags: number theory , Fermat s Little Theorem , IMO

Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

2005 IMO Shortlist, 1

Tags: number theory , Fermat s Little Theorem , IMO

Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

1995 Balkan MO, 3

Tags: modular arithmetic , number theory proposed , number theory , Fermat s Little Theorem

Let $a$ and $b$ be natural numbers with $a > b$ and having the same parity. Prove that the solutions of the equation \[ x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0 \] are natural numbers, none of which is a perfect square. [i]Albania[/i]

2023 Indonesia TST, N

Tags: number theory , Fermat s Little Theorem , primitive root , Primitive Roots

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$