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1^2 ,3^2 ,\ldots,(2n-1)^2 on the blackboard, erasing 3 each time

Source: Balkan BMO Shortlist 2015 N1

August 5, 2019
Squaresnumber theorySum of Squarescombinatorics

Problem Statement

Let dd be an even positive integer. John writes the numbers 12,32,…,(2n−1)21^2 ,3^2 ,\ldots,(2n-1)^2 on the blackboard and then chooses three of them, let them be a1,a2,a3{a_1}, {a_2}, {a_3}, erases them and writes the number 1+∑1≤i<j≤3∣ai−aj∣1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}| He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers 12,32,…,(2n−1)21^2 ,3^2 ,\ldots,(2n-1)^2.
(Albania)
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