Olympiad problems

2016 Saudi Arabia GMO TST, 3

Find all positive integer $n$ such that there exists a permutation $(a_1, a_2,...,a_n)$ of $(1, 2,3,..., n)$ satisfying the condition: $a_1 + a_2 +... + a_k$ is divisible by $k$ for each $k = 1, 2,3,..., n$.

2014 Saudi Arabia GMO TST, 2

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

Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$

2004 Estonia National Olympiad, 4

Tags: divide , divisible , odd , number theory

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

2021 Saudi Arabia BMO TST, 3

Tags: number theory , divide , divisible

Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

2017 Junior Regional Olympiad - FBH, 3

Tags: divisible , number theory

On blackboard there are $10$ different positive integers which sum is equal to $62$. Prove that product of those numbers is divisible with $60$

2020 Israel Olympic Revenge, N

Tags: number theory , divisible

Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers. Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.

2010 Cuba MO, 5

Tags: number theory , divide , 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 $a + p | a^2 + p^2$.

1990 Greece National Olympiad, 3

Tags: divide , divisible , number theory

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

2015 Thailand Mathematical Olympiad, 1

Tags: divide , recurrence relation , divisible , number theory

Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.

1977 Bundeswettbewerb Mathematik, 1

Tags: algebra , number theory , sum , divisible , integer

Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

1925 Eotvos Mathematical Competition, 1

Tags: number theory , divide , divisible

Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

1989 Poland - Second Round, 4

Tags: number theory , divide , divisible

The given integers are $ a_1, a_2, \ldots , a_{11} $ . Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $ such that the number $ x_1a_1 + \ldots x_{11}a_{ 11}$ is divisible by 1989.

2005 Estonia National Olympiad, 2

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

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

2009 Hanoi Open Mathematics Competitions, 5

Tags: number theory , divisible

Prove that $m^7- m$ is divisible by $42$ for any positive integer $m$.

2021 Saudi Arabia Training Tests, 40

Tags: divisible , number theory

Given $m, n$ such that $m > n^{n-1}$ and the number $m+1$, $m+2$,$ ...$, $m+n$ are composite. Prove that there exist distinct primes $p_1, p_2, ..., p_n$ such that $m + k$ is divisible by $p_k$ for each $k = 1, 2, ...$

VMEO IV 2015, 10.3

Tags: number theory , divide , divisible

Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

1986 Tournament Of Towns, (129) 4

Tags: factorial , number theory , divisible , divisor , divide

We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

1999 Tournament Of Towns, 1

Tags: combinatorics , divisible , divide , 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)

1994 Tournament Of Towns, (417) 5

Tags: number theory , digit , divisible , divide

Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

2019 Saudi Arabia BMO TST, 1

Tags: number theory , divide , divisible , prime

Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

2012 NZMOC Camp Selection Problems, 2

Tags: number theory , consecutive , divide , divisible , sum

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

2013 Saudi Arabia Pre-TST, 2.1

Tags: number theory , remainder , divide , divisible

Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.

1957 Moscow Mathematical Olympiad, 357

Tags: number theory , divisible

For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?

2007 Postal Coaching, 6

Tags: digit , number theory , divide , divisible

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

1996 Austrian-Polish Competition, 1

Tags: number theory , divide , divisible , digit

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits