IT-CESENATICO 2013 Problem 1

A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:

[list] (a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$ ;

(b) $v_s,v_p$ and $v_d$ , expressed in meter per second, are integers.

(c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds. [/list]

Find all possible values of $v_s, v_p$ and $v_d$ .