be a convex pentagon with 
and 
For each point 
in the plane, define 
The least possible value of 
can be expressed as 
where 
is not divisible by the square of any prime. Find 
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
.
Y by
Let 
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called 
if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
and 
be positive integers. Let 
be the set of integer points 
with 
and 
. A configuration of 
rectangles is called 
if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
case is trivial. Now, we induct proving that 
is odd if 
is odd.
t![[asy]
unitsize(0.5 cm);
dot((0,0), purple);
dot((1,0), black);
dot((2,0), black);
dot((3,0), purple);
dot((4,0), black);
dot((5,0), black);
dot((0,1), red);
dot((1,1), black);
dot((2,1), black);
dot((3,1), red);
dot((4,1), black);
dot((5,1), black);
dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);
dot((0,3), red);
dot((1,3), black);
dot((2,3), black);
dot((3,3), red);
dot((4,3), black);
dot((5,3), black);
dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);
dot((0,5), purple);
dot((1,5), black);
dot((2,5), black);
dot((3,5), purple);
dot((4,5), black);
dot((5,5),black);
[/asy]](http://latex.artofproblemsolving.com/b/1/4/b147c3592d18385013298e97720574f5e33fe125.png)
ways to pair up the 2n points. From here, there are 
to partition the remaining points into rectangles. Both of these quanitities are odd so total ways in this case are odd.![[asy]
unitsize(0.5 cm);
dot((0,0), purple);
dot((1,0), black);
dot((2,0), purple);
dot((3,0), black);
dot((4,0), black);
dot((5,0), black);
dot((0,1), red);
dot((1,1), black);
dot((2,1), black);
dot((3,1), black);
dot((4,1), red);
dot((5,1), black);
dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);
dot((0,3), red);
dot((1,3), black);
dot((2,3), black);
dot((3,3), black);
dot((4,3), red);
dot((5,3), black);
dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);
dot((0,5), purple);
dot((1,5), black);
dot((2,5), purple);
dot((3,5), black);
dot((4,5), black);
dot((5,5), black);
[/asy]](http://latex.artofproblemsolving.com/5/9/6/5965fa66d20c09cf6a622aebf1bebe9392421668.png)
![[asy]
unitsize(0.5 cm);
dot((0,0), purple);
dot((1,0), black);
dot((2,0), black);
dot((3,0), black);
dot((4,0), purple);
dot((5,0), black);
dot((0,1), red);
dot((1,1), black);
dot((2,1), red);
dot((3,1), black);
dot((4,1), black);
dot((5,1), black);
dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);
dot((0,3), red);
dot((1,3), black);
dot((2,3), red);
dot((3,3), black);
dot((4,3), black);
dot((5,3), black);
dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);
dot((0,5), purple);
dot((1,5), black);
dot((2,5), black);
dot((3,5), black);
dot((4,5), purple);
dot((5,5), black);
[/asy]](http://latex.artofproblemsolving.com/0/1/9/019d37253bfb07489ae46035b82a752909428d06.png)
)
not-necessarily-distinct configurations. The only class of configurations with odd size is the one with 
