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Algebra Level 5

Consider all nn-tuples of real numbers, (x1,x2,x3,...,xn)\displaystyle{(x_{1}, x_{2}, x_{3}, ..., x_{n})} such that i=1nxi2=1\displaystyle{\sum_{i=1}^n x_{i}^{2} =1} where nn is a positive integer.

Let MnM_{n} be the maximum value of i=1nixi\displaystyle{\sum_{i=1}^n i x_{i}} over all such nn-tuples.

Determine the sum of all integers p,1p300\displaystyle{p, 1 \leq p \leq 300} such that MpM_{p} is an integer.

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