[/quote]
,
and 
be the points in which the median and the angle bisector, respectively, at 
meet the side 
meets 
and 
,
the point in which the perpendicular at 
produced.
is perpendicular to 
and 
such that for all 
if 
then 
Find the smallest of sum of all elements of 
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](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
of a prime number 
and a positive integer 
for which 
is an integer.
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
and 
,
and 
meet at 
and 
meet at 
Let 
be the point of intersection of the lines 
and 
be the point of intersection of the lines 
and 
Prove that 
,
.
such that if 
and 
are distinct rational numbers satisfying 
or 
,
?
and 
.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
,
.
real solutions.
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
Y by Adventure10, Mango247, cubres
Let 
be an acute-angled triangle and let 
, 
, and 
be the midpoints of 
, 
, and 
respectively. Construct a circle, centered at the orthocenter of triangle , such that triangle lies in the interior of the circle. Extend 
to intersect the circle at 
, 
to intersect the circle at 
and 
to intersect the circle at 
. Show that 
.
be an acute-angled triangle and let 
, 
, and 
be the midpoints of 
, 
, and 
respectively. Construct a circle, centered at the orthocenter of triangle , such that triangle lies in the interior of the circle. Extend 
to intersect the circle at 
, 
to intersect the circle at 
and 
to intersect the circle at 
. Show that 
.
, so we get 
.
