IMO Shortlist 1969 problem 9
Dodao/la:
arhiva2. travnja 2012. ![(BUL 3)](/media/m/2/a/2/2a26c48217feac1a436c9030a098d2e5.png)
One hundred convex polygons are placed on a square with edge of length
![38 cm.](/media/m/a/4/9/a4937b0e91dff97a99047d1ed370a3de.png)
The area of each of the polygons is smaller than
![\pi cm^2,](/media/m/9/4/1/941587ee96c15fd7c84e9e5fd8fd23e9.png)
and the perimeter of each of the polygons is smaller than
![2\pi cm.](/media/m/b/1/5/b15b7fa513bb98686e377ca1e2cc3ade.png)
Prove that there exists a disk with radius
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
in the square that does not intersect any of the polygons.
%V0
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
Izvor: Međunarodna matematička olimpijada, shortlist 1969