2004 Indonesia MO Problems/Problem 8
Problem 8
A floor with an area of 
will be covered by 
rugs with various shapes, each having an area of 
. Show that there exist 
overlapping rugs with the overlapped area at least 
.
Solution
Let the first 3 rugs occupy the entire floor, then the next rug that you add in, by the pigeon hole principle, it must overlap with another rug. Let a, b and c be the overlapping region with rug 1, rug 2 and rug 3 respectively, a+b+c = 1, thus at least one of a, b and c must be greater than 0.2.
