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min different values of sums of 2 reals out of n

Source: 2022 Czech-Polish-Slovak Match Junior, individual p1 CPSJ

September 3, 2022
combinatoricsalgebra

Problem Statement

Let n≥3n\ge 3. Suppose a1,a2,...,ana_1, a_2, ... , a_n are nn distinct in pairs real numbers. In terms of nn, find the smallest possible number of different assumed values by the following nn numbers: a1+a2,a2+a3,...,an−1+an,an+a1a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1
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