[/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 
.
,
,
,
,
passing through 
and 
at 
and 
and 
are symmetric to 
meets 
and 
at 
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 
are concurrent.
Y by buratinogigle
Let 
be the incentre of 
such that the incircle touches the sides 
at the points 
respectively. The perpendicular to 
at 
and 
meets 
at 
and 
respectively. 
and 
meet the incircle at 
and 
respectively. 
and 
intersect at 
Prove that 
bisects 
be the incentre of 
such that the incircle touches the sides 
at the points 
respectively. The perpendicular to 
at 
and 
meets 
at 
and 
respectively. 
and 
meet the incircle at 
and 
respectively. 
and 
intersect at 
Prove that 
bisects 
,
.
.
.
.
