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Iranian TST 2019, first exam, day1, problem 2

Source:

April 7, 2019

Iranian TSTTSTIran

Problem Statement

a, a_1,a_2,\dots ,a_n

are natural numbers. We know that for any natural number

k

which

ak+1

is square, at least one of

a_1k+1,\dots ,a_n k+1

is also square. Prove

a

is one of

a_1,\dots ,a_n

Proposed by Mohsen Jamali