such that for all positive integers 
,![\[
f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)),
\]](http://latex.artofproblemsolving.com/6/0/7/6071692e2716901e88cc1cfc19ed60388c10701d.png)
where 
and 
denote the least common multiple and the greatest common divisor of 
and 
,
such that 
for all 
.
such that 
coprime, 
and 
.
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](http://latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.
Y by
Given triangle 
inscribed in 
, 
is the midpoint of arc 
of . The perpendicular bisector 
intersects 
at 
. 
intersects 
at 
different from 
. 
is the orthocenter of triangle . Prove that 
= 
inscribed in 
, 
is the midpoint of arc 
of . The perpendicular bisector 
intersects 
at 
. 
intersects 
at 
different from 
. 
is the orthocenter of triangle . Prove that 
= 
.
,
.
is similar to 
,
