MathDB

13

Part of 2002 AIME Problems

Problems(2)

Intermediate Forum AIME Marathon

Source:

8/4/2005
In triangle ABC ABC the medians AD‾ \overline{AD} and CE‾ \overline{CE} have lengths 18 and 27, respectively, and AB \equal{} 24. Extend CE‾ \overline{CE} to intersect the circumcircle of ABC ABC at F F. The area of triangle AFB AFB is mn m\sqrt {n}, where m m and n n are positive integers and n n is not divisible by the square of any prime. Find m \plus{} n.
AMCAIMEgeometrycircumcircleratiogeometric transformationrotation
Area Ratio in Triangle

Source:

12/28/2006
In triangle ABC,ABC, point DD is on BC‾\overline{BC} with CD=2CD=2 and DB=5,DB=5, point EE is on AC‾\overline{AC} with CE=1CE=1 and EA=3,EA=3, AB=8,AB=8, and AD‾\overline{AD} and BE‾\overline{BE} intersect at P.P. Points QQ and RR lie on AB‾\overline{AB} so that PQ‾\overline{PQ} is parallel to CA‾\overline{CA} and PR‾\overline{PR} is parallel to CB‾.\overline{CB}. It is given that the ratio of the area of triangle PQRPQR to the area of triangle ABCABC is m/n,m/n, where mm and nn are relatively prime positive integers. Find m+n.m+n.
geometryratioinequalitiesnumber theoryrelatively primetriangle inequality