3. Three motorcyclists start simultaneously from one point on a circular highway in the same direction. The first motorcyclist caught up with the second for the first time after making 4.5 laps from the start, and 30 minutes before that, he caught up with the third motorcyclist for the first time. The second motorcyclist caught up with the third for the first time three hours after the start. How many laps per hour does the first motorcyclist make?

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# Solution

Let $x, y, z$ be the speeds of the first, second, and third motorcyclists in circles per hour, respectively.

Express the time it takes for the first motorcyclist to catch up with the second, the first to catch up with the third, and the second to catch up with the third in different ways.

We obtain the system of equations

$$
\left\{\begin{array}{c}
\frac{1}{x-y}=\frac{4.5}{x} \\
\frac{1}{x-z}=\frac{4.5}{x}-\frac{1}{2} \\
\frac{1}{y-z}=3
\end{array}\right.
$$

Solving the system of equations, we get the answer $x=3$.

## Grading Criteria

Only the correct answer - 0 points.

Individual expressions for the time of individual motorcyclists are obtained - 1 point.

One, two, or three equations of the system are correctly formulated, or if another method was chosen, depending on the progress in the solution - 2-4 points.

The problem is solved, but contains one or two arithmetic errors, resulting in an incorrect answer - 5-6 points.

The correct answer is obtained with justification - 7 points.