Problem: Eight players entered a round-robin tennis tournament. At the end of the tournament, a player who wins $N$ sets will take home $N^2$ dollars. The entry fee is $17.50 per player. Why is this enough to pay for the prizes?

Intuitive solution: Trying to maximize the total prize, we see that the eight players can take home as much as $7^2 + \ldots + 1^2+0^2=\$140$ dollars. So $\$140\div8 \text{ players}=\$17.50$ as entry fee for each player is enough.

What is a rigorous proof for this problem?


Denote by $x_k$ the number of victories for player $P_k$, and assume $x_1\leq x_2\leq\ldots\leq x_8$. If $x_8<7$ then $P_8$ has lost at least one game against a player $P_k$ with $k<8$. Since $$(x_8+1)^2+(x_k-1)^2=x_8^2+x_k^2+2(x_8-x_k)+2\geq x_8^2+x_k^2+2\ ,$$ the total payout would have been greater when $P_8$ would have won this game as well. It follows that in the case of maximal payout player $P_8$ wins all his games. Therefore we may assume this from now on and start arguing about player $P_7$. If $x_7<6$, then $P_7$ has lost at least one game against a player $P_k$ with $k<7$, and so on.

(Of course this could and should be converted to a full-fledged induction proof.)

It follows that the total payout $W$ is maximal if $x_k=k-1$ $\>(1\leq k\leq 8)$, so that $$W_\max=\sum_{j=1}^7 j^2={7\cdot 8\cdot 15\over 6}=140\ ,$$ which leads to an entrance fee of ${140\over8}=17.50$ dollars.

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