Track / Stage 7 / 5 of 300 #1885 of 2444

Problem 1885

National Olympiad second round; IMO P1/P4

Geometry Difficulty 7.0 Prove it BMO Round 1 · United Kingdom · 2026

Let $ABC$ be an acute-angled triangle with $AB > AC$ . Let $M$ be the midpoint of $BC$ . The circle passing through $M$ that is tangent to $AB$ at $B$ and the circle passing through $M$ that is tangent to $AC$ at $C$ intersect again at $D$ . Prove that $MA \times MD = MB \times MC$ .

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.