## Circles in a square SEEMOUS 2010

Given a square, we consider some circles inside the square such that the sum of lengths of all circles is equal to twice the perimeter of the square.

i) What is the minimum number of circles which satisfy the given hypothesis?

ii) Prove that there exist infinitely many lines which intersect at least three of the given circles.

*SEEMOUS 2010, Problem 2*

* Hint: *This problem has a classical approach. Use the property given, that the sum of the lengths of the circles is twice the perimeter of the square to find the sum of the radii of the given circles. What is the maximum radi for one of our circles? Use that to see how many circles you need. (look at the statement of (ii) to see that the number might be just …)

For the second part, project all circles to one side of the square, and see that the side can be covered more than three times with the respective projections. Choose the points (which exists and are infinitely in number) where at least three projections intersect. Then the orthogonal lines to the side of the square lifted in those points solve our problem.