MathDB

15

Part of 2001 AIME Problems

Problems(2)

Octahedron numbering.

Source: AIME 2001 #15

12/6/2005
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is m/n,m/n, where mm and nn are relatively prime positive integers. Find m+n.m+n.
octahedronprobabilityAMCAIMEsymmetry
Drilling a Tunnel

Source:

12/28/2006
Let EFGHEFGH, EFDCEFDC, and EHBCEHBC be three adjacent square faces of a cube, for which EC=8EC=8, and let AA be the eighth vertex of the cube. Let II, JJ, and KK, be the points on EF‾\overline{EF}, EH‾\overline{EH}, and EC‾\overline{EC}, respectively, so that EI=EJ=EK=2EI=EJ=EK=2. A solid SS is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to AE‾\overline{AE}, and containing the edges, IJ‾\overline{IJ}, JK‾\overline{JK}, and KI‾\overline{KI}. The surface area of SS, including the walls of the tunnel, is m+npm+n\sqrt{p}, where mm, nn, and pp are positive integers and pp is not divisible by the square of any prime. Find m+n+pm+n+p.
geometry3D geometryprismrectangleAMCAIMEanalytic geometry