be a triangle inscribed in a circle 
with orthocenter 
and altitude 
.
be the midpoint of 
.
meets 
.
meets 
.
be the orthogonal projection of 
on the line 
.
meets 
.
.
to 
(including 
(multiplication) between each number. Then he writes the rest of the numbers in the denominator and also writes 
and 
are natural numbers such that
is a prime number. Find all possible values of 
.
denote the set of positive integers. Find all functions 
such that for positive integers 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
be the foot of the altitude from 
to 
,
intersects line 
and 
.
.
be a quadrilateral inscribed in circle 
.
over lines 
and 
and 
.
meets 
meets 
.
.
Y by
Let 
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
