AoPS Academy
be an acute triangle with circumcenter 
. Draw altitude 
, with 
on side 
. The parallel line to 
passing through intersects line 
at point 
. Prove that point and the midpoints of sides 
and are collinear.
+2 w
the product of all 
's positive divisors, where 
. such that ![$\sqrt[2024]{p(n)} \in \mathbb{Z}$](http://latex.artofproblemsolving.com/7/0/4/70442d4ec6ae8b7a3cf9717719e862bf3b5d4a28.png)
and is not a perfect power of another integer.
be a triangle and 
points in the same plane such that is the orthocenter of triangle 
, is the incenter of triangle 
, and is the centroid of triangle 
. If line 
is parallel to 
, prove that 
is the midpoint of segment .
, define
and
meaning 
is the least common multiple of the numbers 
. Determine whether or not there exists a constant 
such that
for every positive integer .
. Let 
and 
be the points on sides and respectively. Circumcircles of 
and 
intersect at the points and 
. Prove that is a point on side if and only if 
