TUYMAADA 2013 S7

Points $A_1$ , $A_2$ , $A_3$ , $A_4$ are the vertices of a regular tetrahedron of edge length $1$ . The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$ , $A_2$ , $A_3$ . Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$ .

A. Kupavsky