# Around The 100-dimensional Unit Sphere

Algebra Level 5

$S=\sum_{k=1}^{99}x_kx_{k+1}$

Find the maximum of $$S$$ when $$\displaystyle \sum_{k=1}^{100}x_k^2=1$$, where $$x_1,\ldots,x_{100}$$ are real numbers. Enter $$\lfloor 10^5 \times S \rfloor$$ as your answer.

Hint: Consider Chebyshev polynomials of the second kind

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