06/19/2020 (2011: Countdown Round, Problem 18)

Q: A circle with diameter 10 units sits atop a square so that the circle is tangent to the square at the center of the top side. The sides of the square are extended until they also are tangent to the circle, as shown. The perimeter of the resulting figure can be written in the form m + nπ units where m and n are integers. What is the value of m + n?

A: 45

The square's perimeter (not including the top side) is 10 * 3 = 30.

The vertical lines' (between the square and the circle) lengths' sum is 10/2 * 2 = 10.

The perimeter of the top half of the circle is 10π * ½ = 5π.

That means m = 30 + 10 = 40 and n = 5. 40 + 5 = 45.