MathDB

15

Part of 2010 AIME Problems

Problems(2)

Congruent Incircles

Source: AIME 2010I Problem 15

3/17/2010
In △ABC \triangle{ABC} with AB=12 AB = 12, BC=13 BC = 13, and AC=15 AC = 15, let M M be a point on AC‾ \overline{AC} such that the incircles of △ABM \triangle{ABM} and △BCM \triangle{BCM} have equal radii. Let p p and q q be positive relatively prime integers such that AMCM=pq \tfrac{AM}{CM} = \tfrac{p}{q}. Find p+q p + q.
geometryratioincentergeometric transformationquadraticstrigonometryhomothety
Two Circumcircles

Source: AIME II 2010 Problem #15

4/1/2010
In triangle ABC ABC, AC \equal{} 13, BC \equal{} 14, and AB\equal{}15. Points M M and D D lie on AC AC with AM\equal{}MC and \angle ABD \equal{} \angle DBC. Points N N and E E lie on AB AB with AN\equal{}NB and \angle ACE \equal{} \angle ECB. Let P P be the point, other than A A, of intersection of the circumcircles of △AMN \triangle AMN and △ADE \triangle ADE. Ray AP AP meets BC BC at Q Q. The ratio BQCQ \frac{BQ}{CQ} can be written in the form mn \frac{m}{n}, where m m and n n are relatively prime positive integers. Find m\minus{}n.
geometrycircumcircleratiogeometric transformationSpiral Similarityangle bisectorAIME