666 Followers Question

Geometry Level 5

\[\sqrt{2}xy+yz+z\sqrt{1-x^2-y^2-z^2}\]

If \(x,y\) and \(z\) are positive reals satisfying \(x^2+y^2+z^2 \leq 1 \), and that the maximum value of the expression above is equal to \(\cos\left(\dfrac \pi n\right) \) for some positive integer \(n\), find \(n\).

Bonus Question: Find the maximal value of the expression below. \[\sqrt{2}x_1x_2+\sum_{k=2}^{n-1}x_kx_{k+1}+x_n \sqrt{1-\sum_{k=1}^{n}x_k^2}\]

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