Question 177
Let $n$ be an odd integer, and let $S$ be a set of $n$ points of the plane with integer coordinates. Consider a permutation $f: S \rightarrow S$ satisfying the following property:
For every pair of points $A$ and $B$ belonging to $S$, the distance between $f(A)$ and $f(B)$ is greater than or equal to the distance between $A$ and $B$.
Show that there exists a point $X$ belonging to $S$ such that $f(X)=X$.
For every pair of points $A$ and $B$ belonging to $S$, the distance between $f(A)$ and $f(B)$ is greater than or equal to the distance between $A$ and $B$.
Show that there exists a point $X$ belonging to $S$ such that $f(X)=X$.