.
,
on side 
.
passing through 
at point 
.
the product of all 
'
.![$\sqrt[2024]{p(n)} \in \mathbb{Z}$](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 parallel to 
is the midpoint of segment 
are there such that 
and 
?
and 
.
be real numbers such that 
Prove that
Let 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
.
if 
.
Y by
In triangle 
, let 
and 
be the midpoints of
, 
and 
, respectively. Show that the triangle is similar to the one formed by its medians 
, 
, 
if and only if one of the sides 
, , is a common tangent to two of the three circles circumscribed to 
, 
, 
, where 
the centroid of .
, let 
and 
be the midpoints of
, 
and 
, respectively. Show that the triangle is similar to the one formed by its medians 
, 
, 
if and only if one of the sides 
, , is a common tangent to two of the three circles circumscribed to 
, 
, 
, where 
the centroid of .
