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Algebra Level 5

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

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

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

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