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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)
such that the reflection of each of these four points about lies on the circle passing through three remaining points.
, the following inequality always holds:![\[
\frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc}
\]](http://latex.artofproblemsolving.com/d/5/8/d582a3269c92041834c02153ee8e946a274e52bd.png)
+1 w
and 
be positive integers. is a divisor of 
and is a divisor of 
.
.
+1 w
with 
, 
, 
. Find point 
in the plane for which 
is minimum.
(plane), 
, where 
and 
is the incenter of 
(it is well-known).
?
, 
and the length 
of the 
-bisector. I'll use remarkable relations : 
![$\begin{array}{cc}
PD/\triangle BPC\ : & (b+c)^2\cdot PD^2=\left(b\cdot PB^2+c\cdot PC^2\right)-a^2bc\ .\\\\
PI/\triangle APD\ : & 4s^2\cdot PI^2=2s\left[a\cdot PA^2+(b+c)\cdot PD^2\right]-\frac {4abcs(s-a)}{b+c}\ .\end{array}$](http://latex.artofproblemsolving.com/6/a/d/6ad5fa8d04da9419829a26dedab5f7835b6befa3.png)
between above relations. Can obtain similarly and the relation 
.
.
is minimum, and compute this minimum.
