AIME 1996 · 第 4 题

AIME 1996 — Problem 4

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

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 $1000x$ .

解析

Solution

$AIME diagram$

(Figure not to scale) The area of the square shadow base is $48 + 1 = 49$ , and so the sides of the shadow are $7$ . Using the similar triangles in blue, $\frac {x}{1} = \frac {1}{6}$ , and $\left\lfloor 1000x \right\rfloor = \boxed{166}$ .

Solution (with more detail)

Let a side of the shadow base not including the side length of cube be $y$ . The triangle that $y$ is part of is similar to the triangle that $x$ is part of on the same side.

By similar triangles, $y = \frac {1}{x}$ . The length of the complete side of the shadow base is $1+\frac {1}{x}$ .

The area of the shadow is $(1+\frac {1}{x})^2-1=48$ .

Solving this quadratic equation, continue as follows in Solution 1.