,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
such that there exist positive integers 
for which the following holds 
for all 
.
is a perfect cube
and 
such that for all 
if 
then 
Find the smallest of sum of all elements of 
Y by HWenslawski, Adventure10
In a triangle 
and 
are points on side 
on side 
, and 
on side 
, such that 
. Let 
be the centroid of triangle 
and 
be the intersection point of 
and 
. Prove that points 
are collinear.
and 
are points on side 
on side 
, and 
on side 
, such that 
. Let 
be the centroid of triangle 
and 
be the intersection point of 
and 
. Prove that points 
are collinear.This post has been edited 1 time. Last edited by parmenides51, Jul 28, 2018, 10:05 PM
Reason: name typo
Reason: name typo
.
.
is a paralellogram with diagonals 
and 
.
be the midpoint of 
.
.
be the centroid of triangle 
.
are collinear, and because 
,
,
.
is also a median of triangle 
.
.
