Olympiad problems

2001 Korea - Final Round, 1

Tags: function , number theory , modular arithmetic , quadratic , relatively prime

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

1983 Vietnam National Olympiad, 1

Tags: modular arithmetic , greatest common divisor , number theory

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

2020 Macedonia Additional BMO TST, 2

Tags: number theory , modular arithmetic , lte lemma

Given are a prime $p$ and a positive integer $a$. Let $q$ be a prime divisor of $\frac{a^{p^3}-1}{a^{p^2}-1}$ and $q\neq p$. Prove that $q\equiv 1 ( \mod p^3)$.

1992 AIME Problems, 11

Tags: geometric transformation , modular arithmetic , rotation , reflection , geometry

Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{19}{92}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.

2022 European Mathematical Cup, 1

Tags: number theory , modular arithmetic , divisibility , divisor

Determine all positive integers $n$ for which there exist positive divisors $a$, $b$, $c$ of $n$ such that $a>b>c$ and $a^2 - b^2$, $b^2 - c^2$, $a^2 - c^2$ are also divisors of $n$.

2009 Hungary-Israel Binational, 1

Tags: relatively prime , algebra , modular arithmetic , polynomial , number theory

For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.

PEN S Problems, 34

Tags: modular arithmetic

Let $S_{n}$ be the sum of the digits of $2^n$. Prove or disprove that $S_{n+1}=S_{n}$ for some positive integer $n$.

2011 AIME Problems, 11

Tags: modular arithmetic , invariant , function

Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.