PK 2016 Q25

P2015 shakes hands with each of the other 2015 candidates (including P2016). In particular he shakes hands with P1, leaving P1 with no more handshakes to perform. P2014 then shakes hands with each of the candidates, not including P1. But then P2 has used up his two handshakes (once with P2015 and once with P2014). Proceeding in this way, we see that P2013 uses up the third shake for P3, P2012 uses up the fourth shake for P4, and so on, until we get towards the half-way point. The handshakes of P1009 again include P2016 and use up the 1007th shake for P1007. By this point, P2016 has shaken hands with each of P1009 to P2015, and now must provide the 1008th shake for P1008, a total of 1008 shakes for P2016.