[/quote]
,
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 
,
has a closed walk of length at most 
passing through each edge of 
at least once.
satisfying the following condition: for any real numbers 
and 
,
is equal to 
or 
contains no term of the form 
and 
.
?
,
,
be a triangle with 
.
,
,
,
,
respectively. A circle 
passing through 
and tangent to 
and 
at 
and 
,
and 
are symmetric to 
meets 
and 
at 
and 
,
meets 
.
.
of acute triangle 
.
intersects 
.
be 
.
,
,
,
,
,
similarly. Prove that 
,
,
are concurrent.
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
grid with each cell colored with an integer from 
to 
.
be a parallelogram with 
A point 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
Y by
Let 
be Rutthee sequence if 
be a positive integer and
for all 
and there are some positive integer 
sastisfied 
Is it possible that there are Rutthee sequence such that there exist positive integer
such that 
and 
also find all possible value of in Rutthee sequence
be Rutthee sequence if 
be a positive integer and
for all 
and there are some positive integer 
sastisfied 
Is it possible that there are Rutthee sequence such that there exist positive integer
such that 
and 
also find all possible value of in Rutthee sequence
