OMCPLP 2015 Problem 4

Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers . Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$ .