HMMT-Nov 2009 guts, Problem 3

Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length 1 . Find the difference between the sum of the areas of the squares and the sum of the areas of the circles.

Solution: The ratio of the area of each square and the circle immediately inside it is $\frac{4}{\pi}$. The total sum of the areas of the squares is $1+\frac{1}{2}+\frac{1}{4}+\ldots=2$. Difference in area is then $2-2 \cdot \frac{4}{\pi}$. $\fbox{2-\frac{\pi}{2}}$.

Answer: 2-\frac{\pi}{2}