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.
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 .
are there such that 
and 
?
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
. Prove that:![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
, 
is the point of intersection of the medians, is the center of the circumscribed circle. It is known that 
and 
. Find the length of the side .
, is the point of intersection of the medians, is the center of the circumscribed circle. It is known that and . Find the length of the side .
and 
.
