In a 8½x11 sheet of paper I drew two equal non-overlapping circles -- both completely inside the paper, of course.
What's the largest portion of the paper I could cover with the circles?
What would be the answer if I drew THREE equal circles?
Two circle case:
Note that we can assume the two circles to be tangent, since if they were not, we could slide them towards their center of gravity while keeping them within the paper. We can then slide one of the circles, while maintaining the point of tangency between the two circles, so that it touches two adjacent edges of the paper. This still keeps them within the paper. Finally, we can revolve the other circle around the fixed circle so that it too is tangent to two adjacent edges of the paper, dilating the two circles if necessary. The end result leads to three possible optimal configurations: two where the two circles share a common tangent edge, and a third where they don't. Computing these configurations yields that the third is the optimal one, with a radius of (39-2sqrt(187))/4 = 2.9126...
(the computation is done easily by using coordinates, using the fact that the two circles pass through the center of the paper by symmetry; details have been omitted)
Two circles case
