2023 AMC 10A Problems/Problem 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 $2023\pi$ ?

[insert asy of diagram below]

Solution

Notice that the area of the shaded region is $(2^2\pi-1^2\pi)+(4^2\pi-3^2\pi)+(6^2\pi-5^2\pi)+ \cdots + (n^2\pi-(n-1)^2pi)$ for any even number $n$ .

Using the difference of squares, this simplifies to $(1+2+3+4+\cdots+n)\pi$ . So, we are basically finding the smallest $n$ such that $\frac{n(n+1)}{2}>2023 \Rightarrow n(n+1)>4046$ . Since $60^2=3600$ , the only option higher than $60$ is $\boxed{\textbf{(E) } 64}$ .

~MrThinker

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2023 AMC 10A Problems/Problem 15

Problem

Solution