Extremes with Equilaterals

Draw the circle with radius $OP$ . By Pompeiu's Theorem, the line segments $OP$ , $OA$ , and $OB$ can make a (possibly degenerate) triangle. Thus, by the triangle inequality, $OA+OB \ge OP$ and $OB+OP \ge OA$ . We know that $OA=20$ and $OB=14$ . Thus, $20+14 \ge OP \implies OP\le 34$ $14+OP\ge 20\implies OP\ge 6$

Thus, $M=34$ and $m=6$ so our answer is $6\times 34=\boxed{204}$ .

Figure

If $OB$ is fixed and $A$ moves along the red circle, then $P$ also moves along a congruent circle (purple). This purple circle is obtained by rotating the red one by $60°$ with center in $B$ ... well in fact is $-60°$

This means that the center $Q$ of this second circle forms an equilateral triangle $OBQ$ .

Given that $OQ= 14$ and the radius of this purple circle is $20$ , is easy to find that $m=6$ and $M=34$ ... and the answers is $204$ .

:)