Question 4
Call a rational number $r$ powerful if $r=p^k/q$ for relatively prime positive integers $p,q$ and an integer $k\gt 1$. Let positive rational numbers $a,b,c$ satisfy $abc=1$. Suppose positive integers $x,y,z$ exist such that $a^x+b^y+c^z$ is an integer. Prove that $a,b,c$ are all powerful.
Proposed by Jeck Lim, Singapore.
Proposed by Jeck Lim, Singapore.