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Nice numbers which are not really nice

Source: Iranian TST 2021, first exam day 2, problem 5

May 17, 2021
Sequencealgebraarithmetic sequence

Problem Statement

Call a triple of numbers Nice if one of them is the average of the other two. Assume that we have 2k+12k+1 distinct real numbers with k2k^2 Nice triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios
Proposed by Morteza Saghafian
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