17.21. We took three numbers $x, y, z$ and computed the absolute values of their pairwise differences $x_{1}=|x-y|, y_{1}=|y-z|$, $z_{1}=|z-x|$. In the same way, from the numbers $x_{1}, y_{1}, z_{1}$ we constructed the numbers $x_{2}, y_{2}, z_{2}$ and so on. It turned out that for some $n$ we got $x_{n}=x, y_{n}=y, z_{n}=z$. Knowing that $x=1$, find $y$ and $z$.
17.21. Answer: $y=z=0$. The numbers $x_{n}, y_{n}, z_{n}$ are non-negative, so the numbers $x, y, z$ are also non-negative. If all the numbers $x$, $y, z$ were positive, then the largest of the numbers $x_{1}, y_{1}, z_{1}$ would be strictly less than the largest of the numbers $x, y, z$, and then the largest of the numbers $x_{n}, y_{n}, z_{n}$ would be strictly less than the largest of the numbers $x, y, z$. Therefore, among the numbers $x, y, z$ there is a 0. Similarly, it is proved that among the numbers $x_{1}, y_{1}, z_{1}$ there is a 0 (for $n=1$ there is nothing to prove, because then $x_{1}=x, y_{1}=y, z_{1}=z$). This means that two of the numbers $x, y, z$ are equal. In the end, we get that the unordered set of numbers $x, y, z$ can be either $0,0,1$ or $0,1,1$. It is easy to check that the second set does not have the required property.

228 Chapter 17. The Pigeonhole Principle. The Rule of the Extreme