A3 In the figure below, the circles have radii 1, 2, 3, 4, and 5. The total area that is contained inside an odd number of these circles is $m \pi$ for a positive number $m$. What is the value of $m$ ?
Solution
A point is inside an odd number of circles if it is in the outermost ring, the third ring, or the middle circle. The area of the middle circle is $\pi$. The third ring is the area contained in the circle of radius 3 but not contained in the circle of radius 2. The area of the third ring is $3^{2} \pi-2^{2} \pi=5 \pi$. The outer ring is the area contained in the circle of radius 5 but not contained in the circle of radius 4 . The area of the fifth ring is $5^{2} \pi-4^{2} \pi=9 \pi$. Thus, the total area is $\pi+5 \pi+9 \pi=15 \pi$, so $m=15$.