MathDB

15

Part of 2009 AIME Problems

Problems(2)

Maximum area

Source: 2009 AIME I #15

3/18/2009
In triangle ABC ABC, AB \equal{} 10, BC \equal{} 14, and CA \equal{} 16. Let D D be a point in the interior of BC‾ \overline{BC}. Let IB I_B and IC I_C denote the incenters of triangles ABD ABD and ACD ACD, respectively. The circumcircles of triangles BIBD BI_BD and CICD CI_CD meet at distinct points P P and D D. The maximum possible area of △BPC \triangle BPC can be expressed in the form a\minus{}b\sqrt{c}, where a a, b b, and c c are positive integers and c c is not divisible by the square of any prime. Find a\plus{}b\plus{}c.
geometryincentercircumcircletrigonometryAMCAIMEperpendicular bisector
Diameter Section Maximization

Source: AIME 2009II Problem 15

4/2/2009
Let MN‾ \overline{MN} be a diameter of a circle with diameter 1 1. Let A A and B B be points on one of the semicircular arcs determined by MN‾ \overline{MN} such that A A is the midpoint of the semicircle and MB\equal{}\frac35. Point C C lies on the other semicircular arc. Let d d be the length of the line segment whose endpoints are the intersections of diameter MN‾ \overline{MN} with the chords AC‾ \overline{AC} and BC‾ \overline{BC}. The largest possible value of d d can be written in the form r\minus{}s\sqrt{t}, where r r, s s, and t t are positive integers and t t is not divisible by the square of any prime. Find r\plus{}s\plus{}t.
trigonometryratioAMCAIME