Problem:
An even number of circles are nested, starting with a radius of 1 and increasing by 1 each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius 2 but outside the circle of radius 1. An example showing 8 circles is displayed below. What is the least number of circles needed to make the total shaded area at least 2023Ο?
Answer Choices:
A. 46
B. 48
C. 56
D. 60
E. 64
Solution:
Suppose there are 2n circles. The shaded area is then
To find the least value of n such that Aβ₯2023Ο, note that nβ22023βββ32. Because 31(2β 31+1)= 1953 and 32(2β 32+1)=2080, the required value is n=32, and there are 2β 32=(E)64β circles.
