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Scary triginequalinometry on the reals

Source: MIPT Undergraduate Competition 2019 1.5 and 2.5

August 17, 2020
inequalitiestrigonometryreal analysiscalculus

Problem Statement

Prove the inequality ∑k=1n(xk−xk−1)2≥4sin⁡2π2n⋅∑k=0nxk2\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2 for any sequence of real numbers x0,x1,...,xnx_0, x_1, ..., x_n for which x0=xn=0.x_0 = x_n = 0.
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