,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
be a convex pentagon such that 
.
is the circumcenter of triangle 
.
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
such that for all 
the following holds:![\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]](http://latex.artofproblemsolving.com/a/c/a/aca098172ea7ce5663e9dd0dd10c2f1495630230.png)
Y by
Given 
. Point 
is the center of the circle passing through the midpoint of BC and the intersections of the perpendiculars from 
and 
with the bisectors of 
and 
. Points 
and 
are defined similarly. Prove that the orthocenter of
is the center of the inscribed circle for .
. Point 
is the center of the circle passing through the midpoint of BC and the intersections of the perpendiculars from 
and 
with the bisectors of 
and 
. Points 
and 
are defined similarly. Prove that the orthocenter of
is the center of the inscribed circle for .
