CN-TST 2020 Problem 1

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$ , and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$ . Prove that if $p>0$ , then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.