Problem:
One hundred concentric circles with radii 1,2,3,β¦,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
==β[(22β12)+(42β32)+(62β52)+β―+(1002β992)]Ο[(2+1)+(4+3)+(6+5)+β―+(100+99)]Ο21ββ
100β
101Οβ
Thus the desired ratio is
21ββ
1002Ο100β
101Οβ=200101β
and m+n=301β.
The problems on this page are the property of the MAA's American Mathematics Competitions