Математическое просвещение

[rsoldo]

A few equal squares with parallel sides cover a figure of area $P$. Prove that we can choose a few of them which doesn't intersect and cover the area $\geq \frac{P}{4}$ .

P.S.This very interesting problem analyzed in - Вторая серия сборников «Математическое просвещение». Nr5/1960, page 146/147.
See: http://www.math.ru
We're waiting nice and explained solution. If you need to do explain the solution given in this magazine (and Russian is ok)
Thx a lot

[Pphantom]

Тема перемещена из форума «Олимпиадные задачи (М)» в форум «Помогите решить / разобраться (М)»
Причина переноса: тематика.

[worm2]

The proof is on page 143 (the very beginning is on page 142). It is based on Blichfeldt's Theorem.
Let the squares have side 1. Consider square lattice with size 2, parallel to squares' sides. According to the Theorem, translate the figure to cover at least $\lceil P/4 \rceil$ points of the lattice. Unit squares that cover different lattice points, cannot intersect each other, so their total area is at least $\lceil P/4 \rceil$.