Pigeonhole principle on two coloured circle Suppose a circle is divided into 200 congruent sectors, with 100 of them coloured red and the other 100 blue. A smaller concentric circle is placed on the larger circle and also so divided and coloured. 
Prove that no matter how the 100 red sectors are chosen for each circle, the smaller circle can be rotated so that at least 100 sectors of the two circles match in colour.

Consider all $200$ rotations of the smaller circle. Through all of these, each sector matches exactly $100,$ so that the total number of matches is $20,000.$ But then one of the rotations must match at least $100$ sectors.
