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number of the subsets of M whose sum of elements equals n

Source: Danube 2018 junior p4

July 22, 2019
combinatoricsSubsetsSets

Problem Statement

Let MM be the set of positive odd integers. For every positive integer nn, denote A(n)A(n) the number of the subsets of MM whose sum of elements equals nn. For instance, A(9)=2A(9) = 2, because there are exactly two subsets of MM with the sum of their elements equal to 99: {9}\{9\} and {1,3,5}\{1, 3, 5\}. a) Prove that A(n)≤A(n+1)A(n) \le A(n + 1) for every integer n≥2n \ge 2. b) Find all the integers n≥2n \ge 2 such that A(n)=A(n+1)A(n) = A(n + 1)
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