Question 76
a) Let $z_1, z_2, z_3, z_4$ be distinct complex numbers of zero sum, having equal absolute values. Prove that the points of affixes $z_1, z_2, z_3, z_4$ are the vertices of a rectangle.
b) Let $x, y, z, t$ be real numbers such that $\sin x + \sin y + \sin z + \sin t = 0$ and $\cos x + \cos y + \cos z + \cos t = 0$. Prove that, for every integer $n$,
$$
\sin(2n + 1)x + \sin(2n + 1)y + \sin(2n + 1)z + \sin(2n + 1)t = 0.
$$
b) Let $x, y, z, t$ be real numbers such that $\sin x + \sin y + \sin z + \sin t = 0$ and $\cos x + \cos y + \cos z + \cos t = 0$. Prove that, for every integer $n$,
$$
\sin(2n + 1)x + \sin(2n + 1)y + \sin(2n + 1)z + \sin(2n + 1)t = 0.
$$