West Bengal RMO 2015 Problem 1 Solution - Triangle Problem

The second stage examination of INMO, the Regional Mathematical Olympiad (RMO) is a three hour examination with six problems. The problems under each topic involve high level of difficulty and sophistication. The book, Challenge and Thrill of Pre-College Mathematics is very useful for preparation of RMO. West Bengal RMO 2015 Problem 1 Solution has been written for RMO preparation series.

Problem:

Two circles $ \Gamma $ and $ \Sigma $, with centers O and O', respectively, are such that O' lies on $ \Gamma $. Let A be a point on $ \Sigma $, and let M be the midpoint of AO'. Let B be another point on $ \Sigma $, such that $ AB~||~OM $. Then prove that the midpoint of AB lies on $ \Gamma $.

Discussion:

Suppose AB intersects $ \Sigma $ at C. Join O'C. Suppose it intersects OM at D. Clearly in $ \Delta AO'C $ M is the midpoint of AO' and DM is parallel to AC. Then D is the midpoint of O'C.

Now O'C is a chord of $ \Gamma $ and we have proved that D is the midpoint of it. Therefore we can say that OD is perpendicular to O'C.

Since AB is parallel to OD (OM), therefore as O'C is perpendicular to OD, therefore O'C is also perpendicular to AB. Since AB is a chord of circle $ \Sigma $ and O'C is a line from center perpendicular to the chord, hence it bisects are chord implying that C is the midpoint of AB.

Chatuspathi:

• What is this topic: Geometry
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, Cheenta Math Olympiad Program, discuss these topics in the ‘Geometry’ module.
• Book Suggestions: Challenge and Thrill of Pre-College Mathematics.
• Paper: RMO 2015 (West Bengal)