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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.