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a_i + a_j = a_k + a_{\ell} between n positiv e integers, with 5 distinct at leas

Source: Singapore Junior Math Olympiad 2010 2nd Round p3 SMO

March 26, 2020

Sumcombinatoricspositive integers

Problem Statement

Let

a_1, a_2, ..., a_n

be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any

1 \le i < j \le n

, there exist

k,\ell

, both different from

i

and

j

such that

a_i + a_j = a_k + a_{\ell}

. What is the smallest possible value of

n