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


Inspiration.

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