AoPS Academy
grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
such that for every pair of distinct positive integers 
, 
is a divisor of 
.
is divisible by 
for every positive integer . that is not a polynomial in (i.e., there does not exist a polynomial ![\(P(X) \in \mathbb{R}[X]\)](http://latex.artofproblemsolving.com/9/8/4/984bd826ad67d4445e27fd211182f8e3d2a13d5f.png)
such that 
for all 
) and satisfies the given condition?
have the Dirichlet propriety, if 
, 
.
with 
subset of 
and have the Dirichlet propriety.
?
?
with the following property: for all positive divisors 
of , we have 
or 
is prime.
. So we try making a linear function of the form 
. But notice![$$r_a + r_b + r_c = 4R \sin \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} + 4R \cos \frac{A}{2} \sin \frac{B}{2} \cos \frac{C}{2} + 4R \cos \frac{A}{2} \cos \frac{B}{2} \sin \frac{C}{2} = 4R [\sin (\frac{A}{2} + \frac{B}{2} + \frac{C}{2}) + \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}] = 4R + r$$](http://latex.artofproblemsolving.com/4/3/1/431b1221f7b43823df6589a25c0c2e1ec0757801.png)
of five variables such that, for any triangle 
with circumradius 
,
,
we have 
?
