Olympiad problems

2024 Regional Competition For Advanced Students, 4

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

1977 Dutch Mathematical Olympiad, 3

Tags: number theory , divide , divisible , multiple

From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.

2017 Latvia Baltic Way TST, 16

Tags: combinatorics , number theory , divide , divisible

Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.

2004 Denmark MO - Mohr Contest, 2

Tags: number theory , divide , divisible

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

2015 JBMO Shortlist, NT1

Tags: number theory , consecutive , divisible , difference , maximum , integer

What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

1999 Tournament Of Towns, 3

Tags: number theory , divide , divisible , sum of squares

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

2007 Switzerland - Final Round, 2

Tags: number theory , divide , divisible

Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.

1941 Moscow Mathematical Olympiad, 072

Tags: number theory , divisible

Find the number $\overline {523abc}$ divisible by $7, 8$ and $9$.

1983 All Soviet Union Mathematical Olympiad, 360

Tags: number theory , divisible , exponential

Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

1982 All Soviet Union Mathematical Olympiad, 329

Tags: number theory , sum of digits , divisible , exponential

a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?

1998 Tournament Of Towns, 3

Tags: combinatorics , divisible , divide

Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.) (G Galperin)

2000 Tournament Of Towns, 5

Tags: consecutive , number theory , divide , divisible

What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ? (S Tokarev)

2017 QEDMO 15th, 9

Tags: divisible , number theory

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 Stanford Mathematics Tournament, 1

Tags: number theory , divide , divisible

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

2015 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible , gcd

Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

2023 Assara - South Russian Girl's MO, 2

Tags: number theory , divisible , divide

The natural numbers $a$ and $b$ are such that $a^a$ is divisible by $b^b$. Can we say that then $a$ is divisible by $b$?

2018 Stars of Mathematics, 2

Tags: perfect square , divisible , number theory

Show that, if $m$ and $n$ are non-zero integers of like parity, and $n^2 -1$ is divisible by $m^2 - n^2 + 1$, then $m^2 - n^2 + 1$ is the square of an integer. Amer. Math. Monthly

2021 Saudi Arabia JBMO TST, 3

Tags: number theory , divisible , divide

We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.

2011 Saudi Arabia Pre-TST, 3.1

Tags: number theory , divide , divisible

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

1982 Poland - Second Round, 5

Tags: number theory , divide , divisible

Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

2009 Chile National Olympiad, 4

Tags: divide , divisible , number theory

Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$

2014 Junior Balkan Team Selection Tests - Moldova, 4

Tags: number theory , divide , divisible

A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.

2015 Caucasus Mathematical Olympiad, 1

Tags: number theory , divisible , sum , digit

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

2011 Regional Olympiad of Mexico Center Zone, 4

Tags: number theory , digit , divisible

Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.

2012 Abels Math Contest (Norwegian MO) Final, 3b

Tags: number theory , divisible , divide

Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?