MathDB

11

Part of 2002 AIME Problems

Problems(2)

Beam of light.

Source:

11/28/2005
Let ABCDABCD and BCFGBCFG be two faces of a cube with AB=12.AB=12. A beam of light emanates from vertex AA and reflects off face BCFGBCFG at point P,P, which is 7 units from BG‾\overline{BG} and 5 units from BC‾.\overline{BC}. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point AA until it next reaches a vertex of the cube is given by mn,m\sqrt{n}, where mm and nn are integers and nn is not divisible by the square of any prime. Find m+n.m+n.
geometry3D geometrygeometric transformationreflectionvectorAMCAIME
Two Infinite Geometric Series

Source:

12/28/2006
Two distinct, real, infinite geometric series each have a sum of 11 and have the same second term. The third term of one of the series is 1/8,1/8, and the second term of both series can be written in the form m−np,\frac{\sqrt{m}-n}{p}, where m,m, n,n, and pp are positive integers and mm is not divisible by the square of any prime. Find 100m+10n+p.100m+10n+p.
ratioAMCAIMESupportgeometric series