Nemotron-IMO-Bench 2026 G37

Let $d_a, d_b, d_c$ be the distances from an interior point $M$ to the sides $BC, CA, AB$ of triangle $ABC$. Let $O$ be the circumcenter of $ABC$, $I$ its incenter and $r$ its inradius. Prove that if $d_ad_bd_c\geq r^3$, then $M$ lies inside or on the circle with center $O$ and radius $OI$.