imc

Question

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 2023? FIGURE

Accepted Answer

Notice that the area of the shaded region is (2^2-1^2)+(4^2-3^2)+(6^2-5^2)+ + (n^2-(n-1)^2 ) for any even number n. Using the difference of squares, this simplifies to (1+2+3+4++n) . So, we are basically finding the smallest n such that n(n+1)2>2023 n(n+1) > 4046. Since 60(61) > 60^2=3600, the only option higher than 60 is 64.