19. Suppose there are a total of 2011 participants in a mathematics competition, and at least 1000 of them are female. Moreover, given any 1011 participants, at least 11 of them are male. How many male participants are there in this competition?
19. Answer: 1011
It is given that there are at least 1000 female participants. Suppose there are more than 1000 female participants. Then we take a group of 1001 female participants, and add any 10 participants to this group of female participants. This will result in a group of 1011 participants with at most 10 male participants, which contradicts the assumption. Therefore, there are exactly 1000 female participants. Hence, the number of male participants is $2011-1000=1011$.