10.1. We added three different two-digit numbers $\overline{x x}, \overline{y y}, \overline{z z}$ and obtained a three-digit number $\overline{x y z}$. Find all possible values of $x, y, z$. The notation $\overline{a b \ldots c}$ means a number where $a$ is the first digit, $b$ is the second digit, $c$ is the last digit.
Answer: $x=1, y=9, z=8$.

Solution. Let's write the equation as $11 x+11 y+11 z=100 x+10 y+$ $+z$. Then $10 z+y=89 x$. If $x=1$, then $10 z+y=89$, and since $z$ and $y$ are digits, we have $z=8, y=9$. If $x \geq 2$, then $10 z+y \geq 178$, which is impossible, as the left side is a two-digit number.