ISL 1979 Problem 36

A regular tetrahedron $A_1B_1C_1D_1$ is inscribed in a regular tetrahedron $ABCD$ , where $A_1$ lies in the plane $BCD$ , $B_1$ in the plane $ACD$ , etc. Prove that $A_1B_1 \ge\frac{ AB}{3}$ .