1996 AIME Problem 4

Question

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is

x

centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube, is

48

square centimeters. Find the greatest integer that does not exceed

1000x.

Answer

Answer: 166

Concepts:cube geometry similarity area

Difficulty rating: 1940

Small hint:

Project the cube’s upper face onto the horizontal surface from the light source

Big hint:

Similar triangles give the projected side length as

\frac{x+1}{x}

Solution:

The light is

x+1

centimeters above the surface and

x

centimeters above the cube’s top face. By similarity, the projection of that unit-square face is a square of side

\frac{x+1}{x}.

This square contains the unit-square area beneath the cube, so

\left(\frac{x+1}{x}\right)^2-1=48.

Therefore

\frac{x+1}{x}=7,

giving

x=\frac{1}{6}.

The greatest integer not exceeding

1000x=166\frac23

is

166.