2002 USAMO 第 1 题

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \le N \le 2^{2002}$. Prove that it is possible to color every subset of $S$ either blue or red so that the following conditions hold:

(a) the union of any two red subsets is red;

(b) the union of any two blue subsets is blue;

(c) there are exactly $N$ red subsets.

令$S$ 为包含2002 个元素的集合，并令$N$ 为$0 \le N \le 2^{2002}$ 的整数。证明可以将 $S$ 的每个子集着色为蓝色或红色，以便满足以下条件：

(a) 任意两个红色子集的并集是红色；

(b) 任意两个蓝色子集的并集是蓝色；

(c) 正好有 $N$ 个红色子集。