### IMO 1961

**Problem B1**
P is inside the triangle ABC. PA intersects BC in D, PB intersects AC in E, and PC intersects AB in F. Prove that at least one of AP/PD, BP/PE, CP/PF does not exceed 2, and at least one is not less than 2.

**Solution**

Take lines through the centroid parallel to the sides of the triangle. The result is then obvious.

Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

3rd IMO 1961

© John Scholes

jscholes@kalva.demon.co.uk

19 Sep 1998

Last corrected/updated 24 Sep 2003