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Two circles (Posted on 2004-10-21) Difficulty: 3 of 5
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?

No Solution Yet Submitted by Federico Kereki    
Rating: 3.2500 (4 votes)

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Solution Three circles case | Comment 3 of 11 |

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
  Posted by David Shin on 2004-10-21 17:25:21

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