3. (7 points) A traveler was riding in a bus and saw a two-digit number on a kilometer post. He fell asleep and woke up an hour later to see a three-digit number on the kilometer post, where the first digit was the same as the second digit an hour ago, the second digit was zero, and the third digit was the same as the first digit an hour ago. Two hours later, he looked out the window and saw a kilometer post with the same number as two hours ago, except the zero was replaced by another digit. Find the speed of the bus (assuming it moved at a constant speed).
Solution. Let the first number the traveler saw be $\overline{x y}=10 x+y$. After an hour, the number became $\overline{y o x}=100 y+x$. After 2 hours, it became $\overline{y z x}=100 x+10 z+x$. Since the bus's speed is constant, $\overline{y o x}-\overline{x y}=\frac{1}{2} \cdot(\overline{y z x}-\overline{y o x})$, that is, $100 y+x-10 x-y=\frac{1}{2} \cdot(100 y+10 z+x-100 y-x), 9 \cdot(11 y-x)=5 z$.

Since the left side is divisible by 9, the right side is also divisible by 9. Therefore, $z=9$, and the speed of the bus is $5 \cdot z$ and equals 45 km/h. Answer. 45 km/h