MathDB

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}.

Problem Statement