Let Our Powers Combine

Algebra Level 5

Find the largest integer \(n\), such that if \( \{ x_i \} _{i=1}^n \) are non-negative real numbers such that

\[ x_1 ^ 6 + x_2 ^ 6 + \ldots + x_n ^ 6 = n, \]

then we will have

\[ x_1 ^2 + x_2 ^ 2 + \ldots + x_n ^ 2 \leq x_1 ^ 5 + x_2 ^ 5 + \ldots + x_n ^ 5. \]

Details and assumptions

The second condition is true for all sets which satisfy the first condition. You are not asked to find just 1 set of numbers which satisfies both conditions.

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