I found a a couple of fun books of problems in geometry, algebra and probability (geometry book here).
This is one of the problems: given the red circles with radius one-half the large black circle, and the blue circle inscribed so as to just fit inside, derive a relation between the radius of the blue circle and the others. This had me scratching my head for a few minutes before the aha moment.
The challenge question is perhaps easier: prove that the filled-in gray area is equal to the area of one of the red circles.
And a hint for the first problem comes from the next graphic, where I've made a copy of the blue circle and positioned it strategically:
