2011 China National Olympiad #6

Let $m,n$ be positive integer numbers. Prove that there exist infinitely many couples of positive integers $(a,b)$ such that \[a+b| am^a+bn^b , \quad\gcd(a,b)=1.\]

Let $p$ a prime number which does not divide $mn$. 

Choose $(a,b)=(k(p-1)^2,k(p-1))$ for any positive integer $k$ :

$m^{k(p-1)^2}\equiv n^{k(p-1)}\equiv 1\pmod p$

$\implies$ $p|(m^{k(p-1)^2}-n^{k(p-1)})$

$\implies$ $kp(p-1)|k(p-1)^2(m^{k(p-1)^2}-n^{k(p-1)})$

$\implies$ $a+b|a(m^a-n^b)$

$\implies$ $a+b|(a+b)n^b+a(m^a-n^b)$

$\implies$ $a+b|am^a+bn^b$

$\square$