Around The 100-dimensional Unit Sphere

Algebra Level 5

S=k=199xkxk+1S=\sum_{k=1}^{99}x_kx_{k+1}

Find the maximum of SS when k=1100xk2=1\displaystyle \sum_{k=1}^{100}x_k^2=1, where x1,,x100x_1,\ldots,x_{100} are real numbers. Enter 105×S \lfloor 10^5 \times S \rfloor as your answer.

Hint: Consider Chebyshev polynomials of the second kind


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