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

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 $\pi\left[(2^2 - 1^2) + (4^2 - 3^2) + \cdots + (100^2 - 99^2)\right] = \pi\left[(2 + 1) + (4 + 3) + \cdots + (100 + 99)\right],$ which is $\pi\,(1 + 2 + \cdots + 100) = 5050\pi.$

The desired ratio is $\frac{5050\pi}{100^2 \pi} = \frac{101}{200},$ so $m + n = 101 + 200 = 301.$

2003 AIME I Problem 2

Solution:

Problem 2 in Other Years