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Sets S s.t. {1,n} in S in {1,2,...,n}

Source: Romanian District Olympiad 2002, Grade X, Problem 4

October 7, 2018
set theorycountingalgebraclosedness

Problem Statement

For any natural number n≥2, n\ge 2, define m(n) m(n) to be the minimum number of elements of a set S S that simultaneously satisfy: \text{(i)}  \{ 1,n\} \subset S\subset \{ 1,2,\ldots ,n\} \text{(ii)}  any element of S, S, distinct from 1, 1, is equal to the sum of two (not necessarily distinct) elements from S. S.
a) Prove that m(n)\ge 1+\left\lfloor \log_2 n \right\rfloor , \forall n\in\mathbb{N}_{\ge 2} . b) Prove that there are infinitely many natural numbers n≥2 n\ge 2 such that m(n)=m(n+1). m(n)=m(n+1).
⌊⌋ \lfloor\rfloor denotes the usual integer part.
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