2008 IMO Shortlist C3

In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \in S$ will be called $k$-friends if there is a point $C \in S$ such that the area of the triangle $A B C$ is equal to $k$. A set $T \subset S$ will be called a $k$-clique if every two points in $T$ are $k$-friends. Find the least positive integer $k$ for which there exists a $k$-clique with more than 200 elements.

在坐标平面中考虑所有具有整数坐标的点的集合$S$。对于正整数 $k$，如果 S$中存在一个点$C，使得三角形 $A B C$ 的面积等于 $k$，则 S$中的两个不同的点$A、B 将称为 $k$ 友元。如果 $T$ 中的每两个点都是 $k$-friends，则集合 $T \subset S$ 将被称为 $k$-clique。找到存在超过 200 个元素的 $k$ 派的最小正整数 $k$。