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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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