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set 1,2,3,4,...,100

Source: Mathematics Regional Olympiad of Mexico Southeast 2015 P6

October 26, 2021
algebra

Problem Statement

If we separate the numbers 1,2,3,4,…,1001,2,3,4,\dots, 100 in two lists with
a1<a2<⋯<a50a_1<a_2<\cdots<a_{50} and b1>b2>⋯>b50b_1>b_2>\cdots>b_{50}
Prove that, no matter how we do the separation,
∣a1−b1∣+∣a2−b2∣+⋯+∣a50−b50∣=2500\vert a_1-b_1\vert +\vert a_2-b_2\vert+\cdots +\vert a_{50}-b_{50}\vert=2500
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