62nd Czech and Slovak Mathematical Olympiad

Substituting ${\cos }^{2}x=a$, ${\cos }^{2}y=b$, ${\cos }^{2}z=c$ leads to the system $$\{\begin{array}{l}1-a+b=\frac{1}{c}-1,\\ 1-b+c=\frac{1}{a}-1,\\ 1-c+a=\frac{1}{b}-1,\end{array}(1)$$ where $a,b,c\in (0,1)$. Adding these equations together yields $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=6,$$ so the harmonic mean of the numbers $a,b,c$ is $\frac{1}{2}$.

Multiplying the equations by the numbers $c,a$, and $b$, respectively, we get $$\begin{array}{rl}c-ac+bc& =1-c,\\ a-ab+ac& =1-a,\\ b-bc+ab& =1-b,\end{array}$$ which sums to $2(a+b+c)=3$. Therefore, the arithmetic mean of the numbers $a,b,c$ is $\frac{1}{2}$, too. Since the arithmetic and harmonic means are equal, it must be that $a=b=c=\frac{1}{2}$. We can easily verify that this triple satisfies the system (1). The solutions of the original system are thus exactly the triples $\left(\frac{1}{4}\pi +\frac{1}{2}k\pi ,\frac{1}{4}\pi +\frac{1}{2}l\pi ,\frac{1}{4}\pi +\frac{1}{2}m\pi \right)$, where $k,l,m$ are integers.

Jiné řešení. We use the substitution from the above solution. The system (1) is cyclic; if a triple $(p,q,r)$ satisfies it, then so do the triples $(q,r,p)$ and $(r,p,q)$. Therefore, it suffices to find the solutions for which $a\ge b$, $a\ge c$, and all the other solutions can then be obtained by cyclic exchange. Let $a\ge b$, $a\ge c$. It follows from the first equation that $1/c=2-a+b\le 2$, so $c\ge \frac{1}{2}$. Similarly, it follows from the third equation that $1/b=2-c+a\ge 2$, so $b\le \frac{1}{2}$, and thus $b\le c$. From the second equation, we have $1/a=2-b+c\ge 2$, so $a\le \frac{1}{2}$. Altogether, $$\frac{1}{2}\ge a\ge c\ge \frac{1}{2},$$ so $a=c=\frac{1}{2}$. Now, any of the equations of the system (1) yields $b=\frac{1}{2}$. Just as in the previous case, we verify that the found triple satisfies the system (1); so the solutions of the system are the triples $\left(\frac{1}{4}\pi +\frac{1}{2}k\pi ,\frac{1}{4}\pi +\frac{1}{2}l\pi ,\frac{1}{4}\pi +\frac{1}{2}m\pi \right)$, where $k,l,m$ are integers.