## 1. Highway in the Basque Country

The Biarritz - Bilbao highway is approximately 150 km long. In my "Mercedes," I drive it 25 minutes faster than my wife in her car. One day, we set off simultaneously: she from Bilbao, and I from Biarritz. When we met, I noticed that the difference in the distances we had left to travel was equal to the difference in time (in minutes) each of us would need to spend on the remaining journey, assuming we both continued at our usual speeds.

At what distance from Bilbao did we meet? How long did we travel before meeting?
1. Let $x$ be the distance from the meeting point to Bilbao. Then the distance to Biarritz is $150-x$. The difference between these distances, $150-2x$, is 25, because according to the problem, the difference in the time intervals required for each of us to reach the final destination is exactly 25 minutes. But if

$$
150-2x=25, \quad \text{then } x=62.5 \text{.}
$$

Next, let the speed of my "Mercedes" be $V$ km/min, and the speed of my wife's car be $v$. The distances we have left to travel after the meeting are $150-62.5=87.5$ km and 62.5 km. From this, we get

$$
62.5 / v = 87.5 / V, \quad 150 / v - 150 / V = 25.
$$

From this system of equations, it follows that

$$
\frac{1}{v} = \frac{1}{V} \cdot \frac{87.5}{62.5} \text{ and } \frac{150}{V} \left( \frac{87.5}{62.5} - 1 \right) = 25
$$

which means the required time is $87.5 / V = 36.46$ min, i.e., approximately 36 min 28 sec.

Then we find: $V = 2.4$ km/min $= 144 \text{ km/h}$ and $v = 1.72 \text{ km/min} = 103 \text{ km/h}$; on a distance of 150 km, this gives a time difference of 25 minutes.