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min no of pair s(i, j) \in S_n x S_n , 1 <| x_i-x_j | <2

Source: OIFMAT II 2012 day 1 p5 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 27, 2021
algebrainequalities

Problem Statement

Let n∈N n \in N . Let's define Sn={1,...,n} S_n = \{1, ..., n \} . Let x1<x2<⋯<xn x_1 <x_2 <\cdots <x_n be any real. Determine the largest possible number of pairs (i,j)∈Sn×Sn (i, j) \in S_n \times S_n with i≠j i \not = j , for which it is true that 1<∣xi−xj∣<2 1 <| x_i-x_j | <2 and justify why said value cannot be higher.
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