MathDB

2

Part of 2000 USAMO

Problems(1)

USAMO 2000 Problem 2

Source:

10/1/2005
Let SS be the set of all triangles ABCABC for which 5(1AP+1BQ+1CR)−3min⁡{AP,BQ,CR}=6r, 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, where rr is the inradius and P,Q,RP, Q, R are the points of tangency of the incircle with sides AB,BC,CA,AB, BC, CA, respectively. Prove that all triangles in SS are isosceles and similar to one another.
AMCUSAMOgeometryinradiusquadratics