MathDB

15

Part of 2005 AIME Problems

Problems(2)

Aime 2005a #15

Source: Aime 2005a #15

11/10/2005
Triangle ABCABC has BC=20BC=20. The incircle of the triangle evenly trisects the median ADAD. If the area of the triangle is mnm \sqrt{n} where mm and nn are integers and nn is not divisible by the square of a prime, find m+nm+n.
AIMEgeometryquadraticsincenterarea of a triangleHeron's formula
Fun Circles

Source: Aime 2005b #15

11/14/2005
Let w1w_{1} and w2w_{2} denote the circles x2+y2+10x−24y−87=0x^{2}+y^{2}+10x-24y-87=0 and x2+y2−10x−24y+153=0x^{2}+y^{2}-10x-24y+153=0, respectively. Let mm be the smallest positive value of aa for which the line y=axy=ax contains the center of a circle that is externally tangent to w2w_{2} and internally tangent to w1w_{1}. Given that m2=p/qm^{2}=p/q, where pp and qq are relatively prime integers, find p+qp+q.
conicsellipseratioanalytic geometrygraphing linesslopetrigonometry