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Maximization of a sum of sines

Source: Czech and Slovak Olympiad 1983, National Round, Problem 1

April 10, 2020
maximizationalgebranational olympiad

Problem Statement

Let nn be a positive integer and k∈[0,n]k\in[0,n] be a fixed real constant. Find the maximum value of ∣∑i=1nsin⁡(2xi)∣\left|\sum_{i=1}^n\sin(2x_i)\right| where x1,…,xnx_1,\ldots,x_n are real numbers satisfying ∑i=1nsin⁡2(xi)=k.\sum_{i=1}^n\sin^2(x_i)=k.
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