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exactly one sequence of 2n + 1 consecutive numbers

Source: Netherlands - Dutch NMO 1973 p2

January 27, 2023
consecutivenumber theorycombinatorics

Problem Statement

Prove that for every nāˆˆNn \in N there exists exactly one sequence of 2n+12n + 1 consecutive numbers, such that the sum of the squares of the first n+1n+1 numbers is equal to the sum of the squares of the last nn numbers. Also express the smallest number of that sequence in terms of nn.
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