13.297. The length of the circular track of the racetrack is $b$ km. Two jockeys, $A$ and $B$, started the race at the same time, and jockey $A$ arrived at the finish line 2 minutes earlier. In another race, jockey $B$ increased his speed by $c$ km/h, while jockey $A$ decreased his speed by $c$ km/h, and as a result, $B$ arrived at the finish line 2 minutes earlier than $A$. Find the speeds of the jockeys in the first race.
Solution.

Let $x, y$ be the speeds of riders $A$ and $B$ in the first race. According to the problem, we have the system $\left\{\begin{array}{l}\frac{b}{y}-\frac{b}{x}=\frac{2}{60}, \\ \frac{b}{x-c}-\frac{b}{y+c}=\frac{2}{60},\end{array}\right.$ from which $x=\frac{c+\sqrt{c^{2}+120 b c}}{2}$ (km/h), $y=\frac{-c+\sqrt{c^{2}+120 b c}}{2}$ (km/h). Answer: $\frac{c+\sqrt{c^{2}+120 b c}}{2}$ and $\frac{-c+\sqrt{c^{2}+120 b c}}{2}$ km/h.