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Inequality on product of differences

Source: Iberoamerican Problem 6

September 22, 2017
inequalitiesIberoamerican

Problem Statement

Let n>2n > 2 be an even positive integer and let a1<a2<⋯<ana_1 < a_2 < \dots < a_n be real numbers such that ak+1−ak≤1a_{k + 1} - a_k \leq 1 for each 1≤k≤n−11 \leq k \leq n - 1. Let AA be the set of ordered pairs (i,j)(i, j) with 1≤i<j≤n1 \leq i < j \leq n such that j−ij - i is even, and let BB the set of ordered pairs (i,j)(i, j) with 1≤i<j≤n1 \leq i < j \leq n such that j−ij - i is odd. Show that
∏(i,j)∈A(aj−ai)>∏(i,j)∈B(aj−ai)\prod_{(i, j) \in A} (a_j - a_i) > \prod_{(i, j) \in B} (a_j - a_i)
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