Olympiad problems

VMEO III 2006 Shortlist, N4

Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.

1911 Eotvos Mathematical Competition, 3

Tags: number theory , divide , divisible

Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.

1984 Bundeswettbewerb Mathematik, 1

Tags: number theory , divisible , divide

The natural numbers $n$ and $z$ are relatively prime and greater than $1$. For $k = 0, 1, 2,..., n - 1$ let $s(k) = 1 + z + z^2 + ...+ z^k.$ Prove that: a) At least one of the numbers $s(k)$ is divisible by $n$. b) If $n$ and $z - 1$ are also coprime, then already one of the numbers $s(k)$ with $k = 0,1, 2,..., n- 2$ is divisible by $n$.

2016 Flanders Math Olympiad, 2

Tags: power , number theory , divide , divisor

Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

1956 Moscow Mathematical Olympiad, 323

Tags: number theory , divisor , divide

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

2020 Swedish Mathematical Competition, 5

Tags: number theory , combinatorics , divide , divisible

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$

1995 Bulgaria National Olympiad, 1

Tags: number theory , divide , divisible

Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

2006 Switzerland - Final Round, 10

Tags: number theory , divisible , divide

Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

2020 Kazakhstan National Olympiad, 1

Tags: number theory , divide

Find all pairs $ (m, n) $ of natural numbers such that $ n ^ 4 \ | \ 2m ^ 5 - 1 $ and $ m ^ 4 \ | \ 2n ^ 5 + 1 $.

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide , sum

Let $n$ be a nonnegative integer and $M =\{n^3, n^3+1, n^3+2, ..., n^3+n\}$. Consider $A$ and $B$ two nonempty, disjoint subsets of $M$ such that the sum of elements of the set $A$ divides the sum of elements of the set $B$. Prove that the number of elements of the set $A$ divides the number of elements of the set $B$.

1956 Polish MO Finals, 4

Tags: number theory , divide , divisible

Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

2002 Estonia National Olympiad, 1

Tags: gcd , lcm , divisible , divide , number theory

The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.