Prove that there are at least 100 pairs of usable boots A store has 200 boots of size A, 200 boots of size B and 200 boots of size C. Among these 600 boots, 300 are of the left foot and 300 are of the right foot. Knowing that usable pairs of boots have the same size and are for different feet, prove that it's possible to find at least 100 pairs of usable boots.

If any size category $A,B,C$ has equal numbers of left and right boots, there are the required $100$ pairs of usable boots and we are done.

Otherwise every size will have a majority of one foot. Since $101\times 3>300$, these cannot all be the same foot, so without loss of generality we can assume that there are two right-foot majorities and one left-foot majority. Now the number of right boots in the right-foot majority sizes is at most $300$, so there are at least $100$ left boots in those two sizes, which can all be matched to right boots, giving us $100$ usable pairs as required.