Very interesting problem.
When two ants A and B meet, they change directions.
Lets view this as when two ants A and B meet, they continue in the same direction but their "original point" gets changed by the reflection property by the radius at the point of intersection as the mirror.
Now, keeping that in mind, we see that in one circle time, an ant will cross another ant at either 0 points or at 2 points diametrically opposite to each other.
By commutative property of reflection for mirrors all crossing through the centre of the circle, after one circle time, all original points will be at their original locations. Hence, all the ants will complete one full circle at one circle time.
Hope that helps.
Thanks
P.S.: I am not really sure how to explain "By commutative property of reflection for mirrors all crossing through the centre of the circle, after one circle time, all original points will be at their original locations." very well.
Pratik
CSE Blog - quant, math, cse puzzles
http://www.pratikpoddarcse.blogspot.com
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