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 $k = 1, 2,3,..., n$.

1999 Israel Grosman Mathematical Olympiad, 1

Tags: divisible , number theory

For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$. Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.

2013 IMAC Arhimede, 2

Tags: number theory , perfect cube , divisible , exponential

For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.

2001 Estonia National Olympiad, 2

Tags: sum , divisible , number theory

Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.

1976 Chisinau City MO, 126

Tags: algebra , polynomial , divisible

Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.

2018 Tuymaada Olympiad, 5

Tags: number theory , divisible , prime number

A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$ is divisible by $p^3$. Prove that $p \leq n+1$. [i]Proposed by Z. Luria[/i]

2008 Tournament Of Towns, 4

Tags: number theory , factorial , divisible

Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.

2003 Estonia National Olympiad, 4

Tags: infinity , number theory , integer , floor function , divisible , divide

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.