[/quote]
,
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
be the median through 
,
at point 
and segment 
meet 
.
be the triangle formed by lines 
be real numbers such that 
Prove that
Where 
Where 
are are the lengths of the sides of a 
g
then![$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$](http://latex.artofproblemsolving.com/9/3/e/93e5529518807b92cd9e0382f5fb07a293128b04.png)
and 
intersect at point 
and 
be a foot of the altitudes from 
and 
and let 
be the midpoint of 
.
cuts 
in half.
and 
.
be an irrational number. Show that for every integer 
,
such that 
.
of the set of 
”
”
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
Y by Adventure10, Mango247
A circle 
is outscribed over an acute triangle 
. 
and 
are altitudes of . The median 
crosses the circle in the point 
. The point 
is chosen on the section such that 
. Prove that the points 
are 
belong to the same circle.
(I.Nagel, Evpatoriya)
is outscribed over an acute triangle 
. 
and 
are altitudes of . The median 
crosses the circle in the point 
. The point 
is chosen on the section such that 
. Prove that the points 
are 
belong to the same circle.(I.Nagel, Evpatoriya)
.
be the orthocenter of 
,
is cyclic. Now, it suffices to show that 
for 
.
.
.
is the diameter of the 
.
and we're done.
(and 
.
is the polar of 
is also the polar of 
since 
form a harmonic range as images of 
.
are on a circle that is the image of the line 
.
