2011 IMO Shortlist C3

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal{S}$ are collinear. By a windmill we mean a process as follows. Start with a line $\ell$ going through a point $P \in \mathcal{S}$. Rotate $\ell$ clockwise around the pivot $P$ until the line contains another point $Q$ of $\mathcal{S}$. The point $Q$ now takes over as the new pivot. This process continues indefinitely, with the pivot always being a point from $\mathcal{S}$. Show that for a suitable $P \in \mathcal{S}$ and a suitable starting line $\ell$ containing $P$, the resulting windmill will visit each point of $\mathcal{S}$ as a pivot infinitely often. ## $\mathrm{C} 4$ Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \ldots, A_{k}$ such that for all integers $n \geq 15$ and all $i \in\{1,2, \ldots, k\}$ there exist two distinct elements of $A_{i}$ whose sum is $n$.

令 $\mathcal{S}$ 为平面上至少两个点的有限集。假设 $\mathcal{S}$ 没有三点共线。我们所说的风车是指如下的过程。从一条穿过点 $P \in \mathcal{S}$ 的线 $\ell$ 开始。围绕枢轴 $P$ 顺时针旋转 $\ell$，直到该线包含 $\mathcal{S}$ 的另一个点 $Q$。点 $Q$ 现在成为新的枢轴。这个过程无限期地继续下去，枢轴始终是 $\mathcal{S}$ 中的一个点。证明对于合适的 $P \in \mathcal{S}$ 和合适的包含 $P$ 的起始线 $\ell$，生成的风车将无限频繁地访问 $\mathcal{S}$ 的每个点作为枢轴。 ## $\mathrm{C} 4$ 确定满足以下性质的最大正整数 $k$：正整数集合可以划分为 $k$ 个子集 $A_{1}, A_{2}, \ldots, A_{k}$，这样对于所有整数 $n \geq 15$ 和所有 $i \in\{1,2, \ldots, k\}$ 存在两个不同的元素$A_{i}$ 其总和为 $n$。