4. Sharik and Matroskin are skiing on a circular track, half of which is an uphill climb and the other half is a downhill descent. Their speeds on the climb are the same and four times slower than their speeds on the descent. The minimum distance by which Sharik lags behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track.
Answer: 24 km. Solution. The minimum lag of Sharik behind Matroskin occurs when Sharik is at the lowest point of the track, and Matroskin is climbing the hill (if Matroskin were descending at this time, it would mean that half of the track is less than 4 km, which is obviously impossible). And it remains until Matroskin reaches the highest point. After that, Sharik climbs another 4 km uphill, while Matroskin descends. If Matroskin had been descending all this time, he would have moved away from Sharik (or rather, from the highest point) by 16 km. But the distance stopped increasing when Matroskin was 13 km ahead of Sharik, which means Matroskin had reached the lowest point of the track and started climbing the hill. Since the lag increased by 3 km for each kilometer Sharik traveled, and it increased by a total of 9 km, Sharik had 4-(13-4)/(4-1) = 1 km left to the highest point when Matroskin reached the lowest point. Therefore, the length of the descent is 13-1 = 12 km, and the entire track is 24 km.