FI-MO 1998

Show that points $A, B, C$ and $D$ can be placed on the plane in such a way that the quadrilateral $ABCD$ has an area which is twice the area of the quadrilateral $ADBC.$

There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks. The committee members have been provided with keys in such a way that every six members can open the safe, but no five members can do that. What is the smallest possible number of locks, and how many keys are needed in that case?

Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$ Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$

There are $110$ points in a unit square. Show that some four of these points reside in a circle whose radius is $1/8.$