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can arithmetic mean of p squares from a set of (p^2 + 1)/2 be an integer?

Source: 2021 Austrian Federal Competition For Advanced Students, Part 2, p6

June 6, 2021
number theoryPerfect SquarePerfect Squares

Problem Statement

Let pp be an odd prime number and MM a set derived from p2+12\frac{p^2 + 1}{2} square numbers. Investigate whether pp elements can be selected from this set whose arithmetic mean is an integer.
(Walther Janous)
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