Olympiad problems

2015 Iran MO (3rd round), 4

$a,b,c,d,k,l$ are positive integers such that for every natural number $n$ the set of prime factors of $n^k+a^n+c,n^l+b^n+d$ are same. prove that $k=l,a=b,c=d$.

1995 Balkan MO, 3

Tags: fermat's little theorem , modular arithmetic , number theory

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]

2020 Macedonian Nationаl Olympiad, 1

Tags: number theory , fermat's little theorem , gcd , divisibility , prime number

Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.

2023 Indonesia TST, N

Tags: number theory , primitive root , fermat's little theorem

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 $LCM(p-1,q-1,r-1)|pqr-1$

2019 USEMO, 4

Tags: number theory , fermat's little theorem

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]

2005 IMO Shortlist, 1

Tags: number theory , fermat's little theorem

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

2023 Indonesia TST, N

Tags: fermat's little theorem , primitive root , number theory

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 $LCM(p-1,q-1,r-1)|pqr-1$

2013 IMAR Test, 1

Tags: prime , number theory , fermat's little theorem

Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.

2024 Baltic Way, 17

Tags: fermat's little theorem , prime number , divisibility , bound , number theory

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 $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

2018 Bulgaria EGMO TST, 2

Tags: number theory , fermat's little theorem , divisibility

Let $m,n \geq 2$ be integers with gcd$(m,n-1) = $gcd$(m,n) = 1$. Prove that among $a_1, a_2, \ldots, a_{m-1}$, where $a_1 = mn+1, a_{k+1} = na_k + 1$, there is at least one composite number.

2012 Regional Olympiad of Mexico Center Zone, 5

Tags: fermat's little theorem , number theory

Consider and odd prime $p$. For each $i$ at $\{1, 2,..., p-1\}$, let $r_i$ be the rest of $i^p$ when it is divided by $p^2$. Find the sum: $r_1 + r_2 + ... + r_{p-1}$

2005 Austrian-Polish Competition, 4

Tags: number theory , perfect square , fermat's little theorem , prime , diophantine equation

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

2005 IMO, 4

Tags: number theory , fermat's little theorem

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

2023 Brazil National Olympiad, 5

Tags: number theory , fermat's little theorem , euler s phi function

An integer $n \geq 3$ is [i]fabulous[/i] when there exists an integer $a$ with $2 \leq a \leq n - 1$ for which $a^n - a$ is divisible by $n$. Find all the [i]fabulous[/i] integers.