[/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.
units horizontally (left or right); or
Y by
An ant is on the Cartesian plane. In a single move, the ant selects a positive integer 
, then either travels
, the ant can choose to go to one of eight possible points.
Prove that, for any integers
and 
, the ant can travel from 
to 
using at most 
moves.
, then either travels
- units vertically (up or down) and
units horizontally (left or right); or - units horizontally (left or right) and units vertically (up or down).
, the ant can choose to go to one of eight possible points.Prove that, for any integers
and 
, the ant can travel from 
to 
using at most 
moves.
in two moves if 
is divisible by 
is also divisible by 
.
horizontally and 
vertically.
horizontally and 
vertically.
is divisible by 
is reachable in two moves, since the values for 
are 
and 
,
