In the city of Cordisburgo, a bicycle race was held on a circular circuit, in which three cyclists, Guimarães, Rosa, and João, participated. In the first hour of the race, Guimarães completed exactly 230 full laps, João completed exactly 111 full laps, but it is not known how many laps Rosa completed. It is only known that the number was an integer and that Rosa completed more laps than João and fewer than Guimarães. Additionally, each of them rode at a constant speed, and they all started from the same point. Considering also the overtakes made at the beginning of the time, how many total overtakes were made in that first hour of the race?
Solution

Let $x$ be the number of laps Rosa completed in the first hour of the race. Since Guimarães completed more laps than Rosa, and both ran at a constant speed, the number of times Guimarães lapped Rosa was

$$
230-x
$$

Since Rosa completed more laps than João, and both ran at a constant speed, the number of times Rosa lapped João was

$$
x-111
$$

Since Guimarães completed more laps than João, and both ran at a constant speed, the number of times Guimarães lapped João was

$$
230-111
$$

Therefore, the total number of overtakes in the first hour of the race was

$$
\begin{aligned}
(230-x)+(x-111)+(230-111) & =230-111+230-111 \\
& =238
\end{aligned}
$$