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 $166.$

1996 AIME Problem 4

Answer: 166

Difficulty rating: 1940

Small Hint:

Big Hint:

Solution:

Problem 4 in Other Years