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.
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 .
such that 
for any relatively prime natural numbers 
.
with 
, are chosen points 
and 
on side , so that
. Segments 
and 
are angle bisectors of 
and 
respectively. Prove that the reflection of in line 
lies on the line .
