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Sum of squares of difference between permutations is even

Source: South African MO 1995 Q3

May 27, 2012
algebra unsolvedalgebra

Problem Statement

Suppose that a1,a2,…,ana_1,a_2,\dots,a_n are the numbers 1,2,3,…,n1,2,3,\dots,n but written in any order. Prove that (a1−1)2+(a2−2)2+⋯+(an−n)2(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2 is always even.
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