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Picking numbers from a set to satisfy congruence

Source: Latvian TST for Baltic Way 2022 P14

November 24, 2022
number theory

Problem Statement

Let AA be a set of 2020 distinct positive integers which are all no greater than 397397. Prove that for any positive integer nn it is possible to pick four (not necessarily distinct) elements x1,x2,x3,x4x_1, x_2, x_3, x_4 of AA satisfying x1≠x2x_1 \neq x_2 and (x1−x2)n≡x3−x4(mod397).(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.
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