Olympiad problems

2018 Turkey MO (2nd Round), 3

A sequence $a_1,a_2,\dots$ satisfy $$ \sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10}, $$ for every $n\in\mathbb{N}$. Let $c$ be a positive integer. Prove that, for every positive integer $n$, $$ \frac{c^{a_n}-c^{a_{n-1}}}{n} $$ is an integer.

2009 China Team Selection Test, 3

Tags: greatest common divisor , number theory , relatively prime , induction , prime number , prime factorization , mobius function

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

2009 China Team Selection Test, 3

Tags: induction , prime factorization , prime number , relatively prime , greatest common divisor , mobius function , number theory

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$