$\underline{\text { Frankin B.R. }}$

Petya marked several (more than two) points on a plane, all distances between which are distinct. A pair of marked points $(A, B)$ is called unusual if $A$ is the farthest marked point from $B$, and $B$ is the nearest marked point to $A$ (excluding the point $A$ itself). What is the maximum possible number of unusual pairs that Petya could have obtained?
Let $(A, B)$ be an unusual pair I, and $K$ be another marked point. Then $B K_{\mathrm{I}} B C>_{\mathrm{II}} C D>_{\mathrm{II}} A D>_{\mathrm{I}} A B$. Contradiction.

An example with one unusual pair $(A, B)$ - the vertices of triangle $A B C$, where $A C>A B>B C$.

## Answer

One pair.