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|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2, diophantine

Source: Singapore Math Olympiad 2017 2nd Round SMO, Junior p2, Senior p1

March 26, 2020

diophantineDiophantine equationnumber theoryfactorial

Problem Statement

Let

n

be a positive integer and

a_1,a_2,...,a_{2n}

2n

distinct integers. Given that the equation

|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2

has an integer solution

x = m

, find

m

in terms of

a_1,a_2,...,a_{2n}