2008 USAMO 第 6 题

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $2^k$ for some positive integer $k$).

在某个数学会议上，每一对数学家要么是朋友，要么是陌生人。用餐时，每个参与者都在两个大餐厅之一用餐。每个数学家都坚持在一个他或她的朋友数量为偶数的房间里吃饭。证明数学家在两个房间之间分配的方式数量是 2 的幂(即，对于某个正整数 $k$，其形式为 $2^k$)。