2023 AMC 10A #15

 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 $2023\pi$?

Solution:

Note that the area of the shaded region for $n$ total circles is $$\pi [(n^2-(n-1{)}^2)+\cdots +(4^2-3^2)+(2^2-1^2)]$$

This must be greater than or equal to $2023\pi$, so we have $$(n^2-(n-1{)}^2)+\dots (4^2-3^2)+(2^2-1^2)\ge 2023$$

For any $k$, $k^2-(k-1{)}^2=2k-1$, so we can write this as

$$⟹(2\cdot 2-1)+(4\cdot 2-1)+\cdots +(2n-1)\ge 2023$$

$$⟹n\ge 64⟶\boxed{{\displaystyle \mathbf{\text{(E) 64}}}}$$