2018 CMO 第 2 题

Let $n$ and $k$ be positive integers and let $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be the length $n$ lattice cube. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red such that if $P$ and $Q$ are red points and $PQ$ is parallel to one of the coordinate axes, then the whole line segment $PQ$ consists of only red points.

Prove that there exists at least $k$ unit cubes of length $1$ , all of whose vertices are colored red.

设 $n$ 和 $k$ 为正整数，并设 $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ 为长度 $n$ 的格子立方体。假设 $T$ 的 $3n^2 - 3n + 1 + k$ 个点被涂成红色，这样，如果 $P$ 和 $Q$ 是红点，并且 $PQ$ 平行于其中一个坐标轴，则整个线段 $PQ$ 仅由红点组成。

证明至少存在 $k$ 个长度为 $1$ 的单位立方体，其所有顶点均为红色。