MathDB

15

Part of 2002 AIME Problems

Problems(2)

Polyhedron

Source: AIME 2002 #15

11/28/2005
Polyhedron ABCDEFGABCDEFG has six faces. Face ABCDABCD is a square with AB=12;AB=12; face ABFGABFG is a trapezoid with AB‾\overline{AB} parallel to GF‾,\overline{GF}, BF=AG=8,BF=AG=8, and GF=6;GF=6; and face CDECDE has CE=DE=14.CE=DE=14. The other three faces are ADEG,BCEF,ADEG, BCEF, and EFG.EFG. The distance from EE to face ABCDABCD is 12. Given that EG2=p−qr,EG^2=p-q\sqrt{r}, where p,q,p, q, and rr are positive integers and rr is not divisible by the square of any prime, find p+q+r.p+q+r.
geometrytrapezoidanalytic geometryvector
Intersecting Circles

Source:

12/28/2006
Circles C1\mathcal{C}_{1} and C2\mathcal{C}_{2} intersect at two points, one of which is (9,6),(9,6), and the product of the radii is 68.68. The x-axis and the line y=mxy=mx, where m>0,m>0, are tangent to both circles. It is given that mm can be written in the form ab/c,a\sqrt{b}/c, where a,a, b,b, and cc are positive integers, bb is not divisible by the square of any prime, and aa and cc are relatively prime. Find a+b+c.a+b+c.
geometrygeometric transformationhomothetyratioanalytic geometryalgebra2002 aime ii 14