This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can't find a solution anywhere online. Here is the problem:
Let $E \subset [0,1]$ be Lebesgue measurable with measure $|E| < 1/2$. Define $$h (s) = \int _ {E^c} \frac{dt}{{(s-t)}^2}$$ where $E^c = [0, 1]\backslash E$. Prove that there exists $t \in E^c$ such that $$\int_E \frac {ds} {h (s) {(s-t)} ^ 2} \leq c {| E |} ^ 2$$ with some constant $c$ that is independent of $E$.
The tricky part of the problem is the quadratic convergence to zero with the measure of $E$. It is easy to see that the left hand side of the inequality is linearly bound with $|E|$. Any ideas (or links to sites with solutions to Miklós Schweitzer problems?)