1. Dima is driving on a straight highway from point A to point B. From point B, a traffic jam is moving towards Dima, and the length of the jam is increasing at a speed of $v$ km/h. The speed of the car in the traffic jam is 10 km/h, and outside the jam - 60 km/h. The navigator in the car shows at any moment how much time is left for Dima to reach point B, based on the length of the traffic jam at that moment. At some point (before reaching the traffic jam), Dima noticed that the navigator shows the same time as it did a few minutes ago. Find $v$.
Answer (in both cases): 12 km/h. Let the time between two identical GPS readings be $t$ hours. During this time, the distance between the car and the traffic jam decreased by $(60+v) t$ km, so the time to cover this distance (from the GPS's perspective) decreased by $(60+v) t / 60$ hours. On the other hand, the traffic jam extended by $vt$ km, and the time to pass through it increased by $vt/10$ hours. These two quantities should be equal, i.e., $(60+v) t=6 v t$, from which $v=12 \mathrm{km} / \mathrm{h}$.