be positive real numbers such that : 
. Prove that : 
.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
.
?
Y by Adventure10, Mango247
A number of spheres with radius 
are being placed in the form of a square pyramid. First, there is a layer in the form of a square with 
spheres. On top of that layer comes the next layer with 
spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.
are being placed in the form of a square pyramid. First, there is a layer in the form of a square with 
spheres. On top of that layer comes the next layer with 
spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.
is 
.
