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existence of sequence of positive integers with pairwise sums

Source: French MO 1996 P5

April 11, 2021
number theorycombinatoricsFrance

Problem Statement

Let nn be a positive integer. We say that a natural number kk has the property CnC_n if there exist 2k2k distinct positive integers a1,b1,…,ak,bka_1,b_1,\ldots,a_k,b_k such that the sums a1+b1,…,ak+bka_1+b_1,\ldots,a_k+b_k are distinct and strictly smaller than nn.
(a) Prove that if kk has the property CnC_n then k≤2n−35k\le \frac{2n-3}{5}. (b) Prove that 55 has the property C14C_{14}. (c) If (2n−3)/5(2n-3)/5 is an integer, prove that it has the property CnC_n.
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