Partition of a rectangle into squares problem
From AoPSWiki
Problem statement
Show that a rectangle can be partitioned into finitely many squares if and only if the ratio of its sides is rational.
Clearly, every rectangle with rational ratio of sides can be partitioned into finitely many equal squares. Thus, the interesting part of the problem is the "only if" one.
Proof via Dirichlet's simultaneous rational approximation theorem
Without loss of generality, we may assume that both sides of our rectange are greater than 
.
We can choose a positive integer 
such that the product of every coordinate of every vertex of every square in our partition becomes almost an integer after multiplication by . So, we may assume that all these coordinates differ from integers by, at most, 
. Now let 
and 
be the horizontal and the vertical side of our rectangle, respectively, and let 
be the sides of the squares in the partition. Denote by 
the nearest integer to 
. Now, draw the horizontal lines at all half-integer heights (i.e., the heights 
) and look at the total length 
of these lines within our rectangle. On one hand, we have 
. (There are 
lines intersecting our rectangle and each of them intersects it by an interval of length .) On the other hand, looking at what happens in each square, we get 
. Thus 
. Similarly, drawing the vertical lines through half-integer points, we arrive at the identity
. Thus 
, i.e., 
.
