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Source: USAMO 2018 P4 and JMO 2018 P5, by Ankan Bhattacharya

April 19, 2018
USA(J)MOUSAMOUSAJMO2018 USAJMO Problem 52018 USAMO Problem 4

Problem Statement

Let pp be a prime, and let a1,…,apa_1, \dots, a_p be integers. Show that there exists an integer kk such that the numbers a1+k,a2+2k,…,ap+pka_1 + k, a_2 + 2k, \dots, a_p + pk produce at least 12p\tfrac{1}{2} p distinct remainders upon division by pp.
Proposed by Ankan Bhattacharya
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