There is a competition involving sixteen people, with each person having 1 round against every other person. If a person wins a round, they get one point. The loser gets zero points, and if it is a draw, both people get half a point. At the end of the rounds, every person ended up with a different total number of points. Prove that there is a person who won more rounds than he had draws using proof by induction or proof by contradiction.
From this problem, I simply deduced that nobody has to draw for everyone to get a different total number of points. Hence, person 1 could get zero points (he lost every round), person 2 could get one point (he won one round but lost every other one), person 3 could get two points (he won two rounds but lost every other one) and so on... However, I think I'm interpreting this question incorrectly as I used neither of the proofs and the question could not possibly be that simple... Any help would be appreciated.
Thanks :)
First of all, we show that one player must have at least $11.5$ points. Otherwise, the sum of the scores would be at most $$3.5+4+4.5+\cdots +11=116$$ which is too small because the sum must be exactly $120$.
We have $15$ rounds. To get at least $11.5$ with $7$ wins, the player needs $9$ draws and with every win less two additional draws are needed, but this would require at least $16$ rounds.
Hence a player having at least $11.5$ points must at least have won $8$ games, completing the proof.