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Distinct sums and consecutive numbers

Source: 2016 Bundeswettbewerb Mathematik Round 2 - #1

September 7, 2016
number theorynumber theory proposedconsecutiveSumSummation

Problem Statement

There are n(n+1)2\tfrac{n(n+1)}{2} distinct sums of two distinct numbers, if there are nn numbers. For which n (n≥3)n \ (n \geq 3) do there exist nn distinct integers, such that those sums are n(n−1)2\tfrac{n(n-1)}{2} consecutive numbers?
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