Question 40
If $a_1, a_2, \dots, a_n$ are $n$ non-zero complex numbers, not necessarily distinct, and $k, l$ are distinct positive integers such that $a_1^k, a_2^k, \dots, a_n^k$ and $a_1^l, a_2^l, \dots, a_n^l$ are two identical collections of numbers. Prove that each $a_j$, $1 \le j \le n$, is a root of unity.