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no of 3-elements subsets of {1,2,...,n} such one is arithmetic mean of other 2

Source: 49th Austrian Mathematical Olympiad Regional Competition (Qualifying Round) 5th April 2018 p3

May 25, 2019

combinatoricsSubsetsarithmetic mean

Problem Statement

Let

n \ge 3

be a natural number. Determine the number

a_n

of all subsets of

\{1, 2,...,n\}

consisting of three elements such that one of them is the arithmetic mean of the other two.
Proposed by Walther Janous