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 
.
be real numbers satisfying
.
then 
.
.
and 
and 
have same sign and 
, then 
under the assumption that the signs alternate otherwise - since only this can give the maximal sum.
is now reduced to the case of 
, and this sum is not influenced by so that one can be assumed to be 0.
that max is 
