13.114. The distance between two cities is $a$ km. Two motorists, setting out from these cities towards each other, will meet halfway if the first one leaves $t$ hours earlier than the second. If, however, they set out towards each other simultaneously, the meeting will occur after $2t$ hours. Determine the speed of each car, assuming that the speeds are constant throughout the journey.
## Solution.

Let $x$ km/h be the speed of the first car, and $y$ km/h be the speed of the second car. The first car will travel half the distance in $\frac{a}{2 x}$ hours; the second car will take $\frac{a}{2 y}$ hours. According to the condition, $\frac{a}{2 x}+t=\frac{a}{2 y}$. In $2 t$ hours, they will travel $2 t(x+y)=a$ km. Solving the system $\left\{\begin{array}{l}2 t(x+y)=a, \\ \frac{a}{2 x}+t=\frac{a}{2 y},\end{array}\right.$ we find $x_{1}=\frac{(\sqrt{5}-1) a}{4 t}, x_{2}=\frac{-a-\sqrt{5} a}{4 t}<0$. Then $y=\frac{(3-\sqrt{5})}{4 t}$.

Answer: $\frac{a(3-\sqrt{5})}{4 t}$ and $\frac{a(\sqrt{5}-1)}{4 t}$ km/h.