10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist?
Solution. Connect points $A$ and $B$ with a segment if the pair $AB$ is isolated, and consider the resulting graph. Notice that it is connected. Indeed, suppose it splits into several connected components; then we find a pair of points from different components with the smallest distance between them. It is easy to see that such a pair is isolated, as any point that falls within their circle and is closer to both of them than they are to each other cannot lie in the same component with both of them. In a connected graph with 2011 vertices, there are at least 2010 edges, thus we have shown that there are at least 2010 isolated pairs. An example is 2011 points lying on a straight line (or on a semicircle).