2011 Putnam B4

In a tournament, 2011 players meet 2011 times to play a
multiplayer game. Every game is played by all 2011 players together and
ends with each of the players either winning or losing. The standings
are kept in two $2011 \times 2011$ matrices, $T = (T_{hk})$ and $W =
(W_{hk})$. Initially,$T=W=0$. After every game, for every$(h,k)$ (including
for $h=k$), if players $h$ and $k$ tied (that is, both won or both lost),
the entry $T_{hk}$ is increased by 1, while if player $h$ won and player $k$
lost, the entry $W_{hk}$ is increased by 1 and $W_{kh}$ is decreased by 1.

Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative

integer divisible by $2^{2010}$.

在一场锦标赛中，2011 年的选手与 2011 年的选手相遇，进行一场比赛

多人游戏。每场比赛均由 2011 年所有球员一起参加，并且

以每个玩家获胜或失败而结束。积分榜

保存在两个 $2011 \times 2011$ 矩阵中，$T = (T_{hk})$ 和 $W =

(W_{hk})$。最初，$T=W=0$。每场比赛结束后，对于每$(h,k)$(包括

对于 $h=k$)，如果玩家 $h$ 和 $k$ 平局(即都赢了或都输了)，

条目 $T_{hk}$ 增加 1，而如果玩家 $h$ 获胜且玩家 $k$

丢失，条目 $W_{hk}$ 增加 1，$W_{kh}$ 减少 1。

证明在锦标赛结束时，$\det(T+iW)$ 是非负数
能被 $2^{2010}$ 整除的整数。