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Number of Solutions mod p

Source: 2004 IrMO Paper 2 Problem 5

December 28, 2017
number theorycombinatorics

Problem Statement

Suppose p,qp,q are distinct primes and SS is a subset of {1,2,…,p−1}\{1,2,\dots ,p-1\}. Let N(S)N(S) denote the number of solutions to the equation ∑i=1qxi≡0mod  p\sum_{i=1}^{q}x_i\equiv 0\mod p where xi∈Sx_i\in S, i=1,2,…,qi=1,2,\dots ,q. Prove that N(S)N(S) is a multiple of qq.
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