RO-NMO 2007 Problem 83

a) In a triangle $MNP$ , the lenghts of the sides are less than $2$ . Prove that the lenght of the altitude corresponding to the side $MN$ is less than $\sqrt {4 - \frac {MN^2}{4}}$ .

b) In a tetrahedron $ABCD$ , at least $5$ edges have their lenghts less than $2$ .Prove that the volume of the tetrahedron is less than $1$ .