2003 AIME I Problem 2

Question

One hundred concentric circles with radii

1,

2,

3,

\ldots,

100

are drawn in a plane. The interior of the circle of radius

1

is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius

100

can be expressed as

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

Answer: 301

Concepts:annulus difference of squares summation

Difficulty rating: 1790

Small hint:

The colors alternate, so the green regions are the annuli between radii

1

and

2,

3

and

4,

and so on

Big hint:

By difference of squares, the green areas total

\pi\,(1 + 2 + 3 + \cdots + 100)

Solution:

The regions alternate red, green, red, green,

\ldots

from the center outward, so the green regions are the annuli between radii

1

and

2,

between

3

and

4,

and so on up to the annulus between

99

and

100.

Their total area is

\begin{aligned} &\scriptsize \pi\left[(2^2 - 1^2) + (4^2 - 3^2) + \cdots + (100^2 - 99^2)\right] \\ &\scriptsize = \pi\left[(2 + 1) + (4 + 3) + \cdots + (100 + 99)\right], \end{aligned}

which is

\pi\,(1 + 2 + \cdots + 100) = 5050\pi.

The desired ratio is

\frac{5050\pi}{100^2 \pi} = \frac{101}{200},

so

m + n = 101 + 200 = 301.