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ASU 044 All Russian MO 1964 8.4 sets of halfs of integers

Source:

June 18, 2019

number theory

Problem Statement

Given an arbitrary set of

2k+1

integers

\{a_1,a_2,...,a_{2k+1}\}

. We make a new set

\{(a_1+a_2)/2, (a_2+a_3)/2, (a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_1)/2\}

and a new one, according to the same rule, and so on... Prove that if we obtain integers only, the initial set consisted of equal integers only.