Olympiad problems

2004 Thailand Mathematical Olympiad, 10

Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.

2002 Estonia National Olympiad, 1

Tags: divisible , gcd , lcm , 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 $n$.

2008 Switzerland - Final Round, 3

Tags: divide , divisible , number theory

Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.

2018 Hanoi Open Mathematics Competitions, 11

Tags: divisible , number theory

Find all positive integers $k$ such that there exists a positive integer $n$, for which $2^n + 11$ is divisible by $2^k - 1$.

1964 Poland - Second Round, 3

Tags: arithmetic progression , arithmetic sequence , divisible , number theory

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

2022 Mediterranean Mathematics Olympiad, 2

Tags: divisible , digit , number theory

(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$. (b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.

1999 Estonia National Olympiad, 1

Tags: prime , odd , divisible , number theory

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

2023 Francophone Mathematical Olympiad, 4

Tags: divide , divisible , number theory

Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$.

2015 Saudi Arabia JBMO TST, 1

Tags: digit , divisible , number theory

A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$

2001 Taiwan National Olympiad, 2

Tags: divisible , number theory

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

2010 Saudi Arabia Pre-TST, 1.3

Tags: divide , digit , divisible , number theory

1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$. 2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.

2017 Switzerland - Final Round, 4

Tags: divide , number theory , divisible

Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$ $$n + 2 | n^p + q^p.$$ Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.

2016 Ecuador NMO (OMEC), 6

Tags: number theory , divisible

A positive integer $n$ is "[i]olympic[/i]" if there are $n$ non-negative integers $x_1, x_2, ..., x_n$ that satisfy that: $\bullet$ There is at least one positive integer $j$: $1 \le j \le n$ such that $x_j \ne 0$. $\bullet$ For any way of choosing $n$ numbers $c_1, c_2, ..., c_n$ from the set $\{-1, 0, 1\}$, where not all $c_i$ are equal to zero, it is true that the sum $c_1x_1 + c_2x_2 +... + c_nx_n$ is not divisible by $n^3$. Find the largest positive "olympic" integer.

2017 QEDMO 15th, 9

Tags: number theory , divisible

Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.

2018 Saudi Arabia GMO TST, 2

Tags: prime , divisible , divide , number theory

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

2021 Auckland Mathematical Olympiad, 4

Tags: number theory , power , divide , divisible

Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

2008 Switzerland - Final Round, 6

Tags: odd , divide , divisible , number theory

Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

1908 Eotvos Mathematical Competition, 1

Tags: divide , number theory , divisible

Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.

2013 QEDMO 13th or 12th, 2

Tags: number theory , binomial , divide , divisible

Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

2016 Saudi Arabia Pre-TST, 2.4

Tags: divisible , number theory , product , divide

Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

1995 Poland - Second Round, 1

Tags: integer polynomial , polynomial , divisible , algebra

For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.

2008 Abels Math Contest (Norwegian MO) Final, 1

Tags: divisible , integer , number theory

Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$. (a) Show that $s(n)$ is an integer whenever $n$ is an integer. (b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?

2001 Austria Beginners' Competition, 1

Tags: number theory , divide , divisible

Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

2004 All-Russian Olympiad Regional Round, 9.5

Tags: combinatorics , divisible , number theory

The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?

2007 Junior Tuymaada Olympiad, 3

Tags: divisible , combinatorial geometry , number theory , combinatorics

A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure: In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.