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(a_{i1} -a_1)... (a_{i5} -a_5) always even Puerto Rico OMCC TST 2018.2

Source:

September 16, 2021

combinatoricspermutationspermutation

Problem Statement

Let

A = \{a_1, a_2, a_3, a_4, a_5\}

be a set of

5

positive integers. Show that for any rearrangement of

A

a_{i1}

a_{i2}

a_{i3}

a_{i4}

a_{i5}

, the product

(a_{i1} -a_1) (a_{i2} -a_2) (a_{i3} -a_3) (a_{i4} -a_4) (a_{i5} -a_5)

is always even.