3. Two motorcyclists are moving on a circular path that is $1650 \mathrm{~m}$ long at constant speeds. If the motorcyclists move in opposite directions, they meet every minute, and if they move in the same direction, the motorcyclist with the higher speed catches up to the other motorcyclist every eleven minutes. Determine the speeds of the motorcyclists.
Solution. Let the faster motorcyclist travel $x$ meters per minute, and the slower one travel $y$ meters per minute. From the first condition, we have $x+y=1650$, and from the second condition, $x-y=\frac{1650}{11}=150$. The solution to the obtained system of equations is $x=900 \mathrm{~m} / \mathrm{min}$ and $y=700 \mathrm{~m} / \mathrm{min}$. Therefore, the first motorcyclist traveled at a speed of $54 \mathrm{~km} / \mathrm{h}$, and the second one at a speed of $45 \mathrm{~km} / \mathrm{h}$.