MathDB

5

Part of 2018 USAJMO

Problems(1)

Sad Combinatorics

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

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