Question 15
Let $n$ be a positive integer. Prove that there is only one tuple of $n+2$ real numbers $(\alpha_1,\alpha_2,\alpha_3,\ldots,\alpha_n,\alpha_{n+1})$ such that $\alpha_0=\alpha_{n+1}=0$, $|\alpha_i|\le\pi/6$, and
$$1+\sin\alpha_{i+1}+3\sin\alpha_{i-1}=10\sin\alpha_i-2\sin3\alpha_i$$
for every $i=1,2,\ldots,n$.
$$1+\sin\alpha_{i+1}+3\sin\alpha_{i-1}=10\sin\alpha_i-2\sin3\alpha_i$$
for every $i=1,2,\ldots,n$.