This problem was given without answer in a book by Vinogradov (Fundamentals of number theory). I find it nice and maybe useful for beginners because it raises modulo $p$ an analogy with the well-known Vieta's formula in elemental algebra.
Assume that $$a_0x^n+a_1x^{n-1}+\cdots+a_n\equiv0\pmod p$$ admits n solutions.
Prove that $$a_i\equiv (-1)^i a_0S_i; 1\le i\le n$$ where the $S_i$'s are the Vieta's coefficients (the elementary symmetric polynomials)