Olympiad problems

2015 Puerto Rico Team Selection Test, 8

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

2007 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory , divisible , divides

Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

2017 Puerto Rico Team Selection Test, 3

Tags: number theory , multiple , divides

Given are $n$ integers. Prove that at least one of the following conditions applies: 1) One of the numbers is a multiple of $n$. 2) You can choose $k\le n$ numbers whose sum is a multiple of $ n$.

1987 Tournament Of Towns, (159) 3

Tags: number theory , divisible , divides

Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .

2005 Thailand Mathematical Olympiad, 2

Tags: number theory , divides , divisible

Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.

1999 Ukraine Team Selection Test, 8

Tags: divides , number theory

Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.

1998 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divides

Let $X$ be a finite set of positive integers. Prove that for every subset $A$ of $X$, there is a subset $B$ of $X$, with the following property: For each element $ e$ of $X$, $e$ divides an odd number of elements of $B$, if and only if $e$ is an element of $A$.

2016 Dutch BxMO TST, 5

Tags: number theory , divides

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divides , divisible , powers

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

2010 Cuba MO, 5

Tags: number theory , divides , divisible

Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.

2019 Costa Rica - Final Round, 5

Tags: number theory , divides

We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

2011 Saudi Arabia Pre-TST, 3.1

Tags: number theory , divides , divisible

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

2014 Czech-Polish-Slovak Junior Match, 5

Tags: game , number theory , divides , divisor

There is the number $1$ on the board at the beginning. If the number $a$ is written on the board, then we can also write a natural number $b$ such that $a + b + 1$ is a divisor of $a^2 + b^2 + 1$. Can any positive integer appear on the board after a certain time? Justify your answer.

1999 Tournament Of Towns, 1

Tags: combinatorics , divisible , divides , Sum

For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order? (A Shapovalov)

2018 Singapore Junior Math Olympiad, 1

Tags: multiple , number theory , divides , Digits

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

2003 Austria Beginners' Competition, 3

Tags: number theory , consecutive , divides , divisible

a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

1908 Eotvos Mathematical Competition, 1

Tags: number theory , divides , 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 $2^n$.

2021 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , divisible , divides

Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

2013 Czech-Polish-Slovak Junior Match, 4

Tags: divides , divisor , number theory , Digits

Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

2006 QEDMO 2nd, 5

Tags: number theory , divides

For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)

1993 All-Russian Olympiad Regional Round, 11.2

Tags: number theory , divides , divisible

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

2015 Denmark MO - Mohr Contest, 2

Tags: number theory , divides

The numbers $1, 2, 3, . . . , 624$ are paired in such a way that the sum of the two numbers in each pair is $625$. For example $1$ and $624$ form a pair, and $30$ and $595$ form a pair. In how many of the $312$ pairs does the smaller number evenly divide the larger?

2013 India PRMO, 13

Tags: number theory , maximum , divides

To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?

2012 China Northern MO, 8

Tags: number theory , divides

Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that : (1) $(a+1, \frac{a^p+1}{a+1}) = p$ (2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$. (3) $p!|(a +1) $.

2016 Grand Duchy of Lithuania, 4

Tags: divides , divisible , number theory

Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.