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?
Not a complete proof:
I'm assuming the optimal configuration is where one circle A is tangent to one of the long sides at its midpoint, and where each of the other two circles B and C are tangent to this circle, a short side, and the opposite long side.
With this assumption, the radius is uniquely determined: simply look at the right triangle formed by the centers of A and B and the midpoint of the centers of B and C. The hypotenuse is 2r, and the two side lengths are 8.5-2r and 11/2-r. Applying the Pythagorean theorem (and assuming I didn't make a mistake in my computations), we get r=(45-sqrt(1615))/2=2.4064...
I have some ideas of how to justify the assumption, though things might get a bit hairy. Lots of case analysis should do the trick, I think...
Edited on October 21, 2004, 5:26 pm