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Linear combinations of n-th roots of unity cannot vanish so many

Source: China Additional TST for IMO 2020, P1

October 20, 2020
complex numbersalgebraroots of unity

Problem Statement

Let ω\omega be a nn -th primitive root of unity. Given complex numbers a1,a2,⋯ ,ana_1,a_2,\cdots,a_n, and pp of them are non-zero. Let bk=∑i=1naiωkib_k=\sum_{i=1}^n a_i \omega^{ki} for k=1,2,⋯ ,nk=1,2,\cdots, n. Prove that if p>0p>0, then at least np\tfrac{n}{p} numbers in b1,b2,⋯ ,bnb_1,b_2,\cdots,b_n are non-zero.
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