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Fantastic Triplets

Source: Canadian Repêchage 2012: Problem 3

May 19, 2014
geometric sequencenumber theory proposednumber theory

Problem Statement

We say that (a,b,c)(a,b,c) form a fantastic triplet if a,b,ca,b,c are positive integers, a,b,ca,b,c form a geometric sequence, and a,b+1,ca,b+1,c form an arithmetic sequence. For example, (2,4,8)(2,4,8) and (8,12,18)(8,12,18) are fantastic triplets. Prove that there exist infinitely many fantastic triplets.
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