Solutions to $\cos(\pi x) + \cos(\pi y) + \cos(\pi z) = 0$ for rational $x$, $y$, $z$ with $0 \leq x \leq \frac{1}{2} \leq y \leq z \leq 1$

Question

I saw the following equation in a problems book, but it gave no solutions, and I have no idea how to solve it.

Let $x,y,z$ be three rational numbers such that $0 \leq x \leq \frac{1}{2} \leq y \leq z \leq 1$. The equation $$\cos(\pi x) + \cos(\pi y) + \cos(\pi z) = 0$$ has three sets of solutions:

1. $x$ arbitrary, $y = 1/2$, $z = 1 - x$;
2. $x$ arbitrary, $y = 2/3 - x$, $z = 2/3 + x$;
3. $x = 1/5, y = 3/5, z = 2/3$.

Show that these are the only rational solutions.

Answer 1

I found the source of this problem. It's in Acta Arithmetica XXX (1976) 229-240.

http://matwbn.icm.edu.pl/ksiazki/aa/aa30/aa3033.pdf