1. From point $A$ of a circular track, a car and a motorcycle started simultaneously and in the same direction. The car drove two laps without stopping in one direction. At the moment when the car caught up with the motorcyclist, the motorcyclist turned around and increased his speed by $16 \mathrm{~km} / \mathrm{u}$, and after $3 / 8 \mathrm{~h}$ after the turn, arrived at point $A$ simultaneously with the car. Find the entire distance (in km) traveled by the motorcyclist, if this distance is 5.25 km shorter than the entire track. Provide the answer as a number without specifying the unit.

(5 points)
Solution. Let $x$ (km/h) be the speed of the motorcyclist, $y$ (km/h) be the speed of the car, and $S$ (km) be the distance the motorcyclist travels before turning around. Then the total length of the track is $2 S + 5.25$. We have $\frac{S}{x} = \frac{3 S + 5.25}{y}$, $\frac{3 x}{8} + 6 = S$, $\frac{3 y}{8} = S + 5.25$. This leads to the quadratic equation $4 S^2 - 36 S - 63 = 0$, whose positive solution is $S = 10.5$, and the entire distance traveled by the motorcyclist is $2 S = 21$.

Answer: 21.