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Polish MO Final 2010, 5th problem (remainder of the product)

Source:

November 7, 2010
modular arithmeticquadraticsOnline Math Opentrigonometrynumber theorynumber theory proposed

Problem Statement

Prime number p>3p>3 is congruent to 22 modulo 33. Let ak=k2+k+1a_k = k^2 + k +1 for k=1,2,…,p−1k=1, 2, \ldots, p-1. Prove that product a1a2…ap−1a_1a_2\ldots a_{p-1} is congruent to 33 modulo pp.
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