A Graph Theory Olympiad Question Whose Answer is 1015056

April’s Math Olympiad Question was a particularly tough one, only four people in the world solved it! One from Japan, one from Slovakia, one from Ankara, and one from Singapore!

The question starts off seemingly simple enough:

In a party attended by 2015 guests among any 7 guests at most 12 handshakes had been
exchanged. Determine the maximal possible total number of handshakes.

However, when one starts trying out the questions, one quickly realizes the number of handshakes is very large, possibly even up to millions. This question definitely can’t be solved by trial and error!

This question is ideally modeled by a graph, and has connections to the idea of a Turán graph.

The official solution can be accessed here: http://www.fen.bilkent.edu.tr/~cvmath/Problem/1504a.pdf


Turan 13-4.svg
The Turán graph T(13,4)


To read more about Math Olympiad books, you may check out my earlier post on Recommended Math Olympiad books for self-learning.