MathDB

We have

n

guinea pigs placed on the vertices of a regular polygon with

n

sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between

1 km/h

2km/h

,..., and

n km/h

, each one with a distinct velocity, and each guinea pig has a counter starting from

0

. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each

1\leq i\leq n

we have that the

i-

th guinea pig has

i-1

in its counter.

Problem Statement