The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space?
Alexey Tolpygo
a) Let it false. Then exists square with side $=3$ that contains all 5 points. Divide this square by midlines to $4$ squares with side $\frac{3}{2}$. One of such square contains at least $2$ points, but maximal distance in this square can be only $\frac{3}{2} \sqrt{2} = \sqrt{ \frac{9}{2}}>2$
b) Take equilateral square pyramid