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IMO Shortlist 2011, Number Theory 7

Source: IMO Shortlist 2011, Number Theory 7

July 11, 2012
modular arithmeticnumber theoryIMO Shortlist

Problem Statement

Let pp be an odd prime number. For every integer a,a, define the number Sa=∑j=1p−1ajj.S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}. Let m,n∈Z,m,n \in \mathbb{Z}, such that S3+S4−3S2=mn.S_3 + S_4 - 3S_2 = \frac{m}{n}. Prove that pp divides m.m.
Proposed by Romeo Meštrović, Montenegro
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