RO-NMO 2005 Problem 101

Let $n$ be a positive integer, $n\geq 2$ . For each $t\in \mathbb{R}$ , $t\neq k\pi$ , $k\in\mathbb{Z}$ , we consider the numbers $$x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}.$$ Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$ .

Constantin Buse