12

Part of 2001 AIME Problems

Problems(2)

Inscribed Sphere

Source: AIME 2001 #12

A sphere is inscribed in the tetrahedron whose vertices are

A=(6,0,0), B=(0,4,0), C=(0,0,2),

D=(0,0,0).

The radius of the sphere is

m/n,

m

n

are relatively prime positive integers. Find

m+n.

geometry3D geometryspheretetrahedroninradiuspyramidincenter

Sequence of Polyhedra

Source:

Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra

P_{i}

is defined recursively as follows:

P_{0}

is a regular tetrahedron whose volume is 1. To obtain

P_{i+1}

, replace the midpoint triangle of every face of

P_{i}

by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of

P_{3}

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m+n

geometry3D geometrytetrahedronnumber theoryrelatively prime