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iran TST

Source: iran TST 2015 third exam p3

June 6, 2015
combinatorics

Problem Statement

a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n are 2n2n positive real numbers such that a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n aren't all equal. And assume that we can divide a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n into two subsets with equal sums.similarly b1,b2,⋯ ,bnb_1,b_2,\cdots ,b_n have these two conditions. Prove that there exist a simple 2n2n-gon with sides a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n and parallel to coordinate axises Such that the lengths of horizontal sides are among a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n and the lengths of vertical sides are among b1,b2,⋯ ,bnb_1,b_2,\cdots ,b_n.(simple polygon is a polygon such that it doesn't intersect itself)
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