be reals such that 
and 
.
be the side lengths of a triangle. Prove that![\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]](http://latex.artofproblemsolving.com/f/8/0/f80aed170f4f4bd78a2e881ce93b6b7ddfa03c22.png)
be a triangle with 
its orthocenter. The circle with diameter ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
cuts the circumcircle of triangle 
at 
.
and 
is the midpoint of the segment ![$[BH]$](http://latex.artofproblemsolving.com/a/9/f/a9fd89d48264ff1c3f9b77f62ef66f916978d99d.png)
of positive integers is called central if for every positive integer 
,
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
Y by Adventure10, Mango247
Point 
is the reflection of incentre 
of triangle 
across the side 
. The circumcircle of 
intersects the line 
again at point 
. It is known that lies outside the incircle of the triangle . Two tangents drawn from to the latter circle touch it at points 
and 
. Prove that the line 
contains a medial line of the triangle .
Author: L. Emelyanov
is the reflection of incentre 
of triangle 
across the side 
. The circumcircle of 
intersects the line 
again at point 
. It is known that lies outside the incircle of the triangle . Two tangents drawn from to the latter circle touch it at points 
and 
. Prove that the line 
contains a medial line of the triangle .Author: L. Emelyanov
This post has been edited 1 time. Last edited by pohoatza, Jul 17, 2008, 1:02 PM
,
,
be the tangency points of the incircle 
with the sides 
and 
,
be the midpoint of small arc 
.
,
of 
.
is the 
and the reflection 
of 
and 
is the polar of 
,
and 
are concurrent. On the other hand, since 
and so 
.
,
is incircle's radius.
is cyclic and 
,![\[ \boxed{PD=\frac{(p-b)(p-c)}{r}}
\]](http://latex.artofproblemsolving.com/c/a/1/ca1e0b565d290dd0547096441db67cc7986cd3aa.png)
.
and 
.![\[ \boxed{PZ=\frac{\frac{(p-b)(p-c)}{r}\left(\frac{(p-b)(p-c)}{r}-2r\right)}{\frac{(p-b)(p-c)}{r}-r}}
\]](http://latex.artofproblemsolving.com/c/1/a/c1aee31113f6755ed110c8113341dfb2122237e5.png)
and it is enough to show that 
,
is the pole of 
,
.
be the inradius and 
be the 
-
be the 
is a parallelogram, so 
.
be the inverse of 
in the incircle. We have 
Clearly, the polar of 
which is the 
