MathDB

11

Part of 2015 AIME Problems

Problems(2)

Angle Bisector in Isosceles Triangle

Source: 2015 AIME I Problem 11

3/20/2015
Triangle ABCABC has positive integer side lengths with AB=ACAB=AC. Let II be the intersection of the bisectors of ∠B\angle B and ∠C\angle C. Suppose BI=8BI=8. Find the smallest possible perimeter of △ABC\triangle ABC.
geometryangle bisectorAMCAIMEAIME I
Circle Circumscribing a Triangle

Source: 2015 AIME 2 Problem 11

3/26/2015
The circumcircle of acute △ABC\triangle ABC has center OO. The line passing through point OO perpendicular to OB‾\overline{OB} intersects lines ABAB and BCBC at PP and QQ, respectively. Also AB=5AB=5, BC=4BC=4, BQ=4.5BQ=4.5, and BP=mnBP=\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
circumcirclegeometrygeometry solved