BdMO Online Forum

Call a rational number $r$ powerful if $r$ can be expressed in the form $\dfrac{p^k}{q}$ for some relatively prime positive integers $p,q$ and some integer $k > 1$. Let $a,b,c$ be positive rational numbers such that $abc = 1$. Suppose there exist positive integers $x,y,z$ such that $a^x + b^y + c^z$ is an integer. Prove that $a,b,c$ are all powerful.