Problem:
One hundred concentric circles with radii 1,2,3,…,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 m/n, where m and n are relatively prime positive integers. Find m+n.
Solution:
The sum of the areas of the green regions is
==​[(22−12)+(42−32)+(62−52)+⋯+(1002−992)]π[(2+1)+(4+3)+(6+5)+⋯+(100+99)]π21​⋅100⋅101π​
Thus the desired ratio is
21​⋅1002π100⋅101π​=200101​
and m+n=301​.
The problems on this page are the property of the MAA's American Mathematics Competitions