If no two people are to reach the same handshake total, then the most gregarious guy in the stadium must shake 49,999 hands, the next-most-gregarious 49,998, and so on. But that means the shyest guy would shake zero hands — which is impossible, since the gregarious guy shook every hand in the place.
Alternatively, if the shyest guy shakes one hand, then to avoid any duplication the gregarious guy would have to shake 50,000 hands, which is also impossible, as there are only 49,999 hands available for him to shake. Therefore there must be at least two participants, somewhere, who have shaken the same number of hands.
