2007 IMO Shortlist C5

In the Cartesian coordinate plane define the strip $S_{n}=\{(x, y) \mid n \leq x<n+1\}$ for every integer $n$. Assume that each strip $S_{n}$ is colored either red or blue, and let $a$ and $b$ be two distinct positive integers. Prove that there exists a rectangle with side lengths $a$ and $b$ such that its vertices have the same color. (Romania)

在笛卡尔坐标平面中，为每个整数 $n$ 定义条带 $S_{n}=\{(x, y) \mid n \leq x<n+1\}$。假设每个条带 $S_{n}$ 的颜色为红色或蓝色，并令 $a$ 和 $b$ 为两个不同的正整数。证明存在一个边长为 $a$ 和 $b$ 的矩形，且其顶点颜色相同。 (罗马尼亚)