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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
, such that there are 
points 
, 
, 
, 
, 
in the plane with 
that satisfy the following condition: for each 
triangles 
and 
are equal.
or 
is equal to 
, thus 
is on one of the circles of radius and centre or . In the same way is on one of the circles of radius 
with centre or . The intersection of these four circles has no more than 8 points so that 
.
with centre and radius intersects circle 
with centre and radius in two points 
and 
which satisfy the conditions of the problem. Then in triangles 
and 
we have 
and 
. Since these triangles are congruent then 
, therefore and are on the perpendicular bisector of 
. On the other hand 
is perpendicular to segment 
. Then 
and and are the diagonals or non-parallel sides of a trapezoid.
. Then 
. It follows that the distance from to the perpendicular bisector of must be less than the distance from to this line otherwise we obtain a contradiction to the condition . Then for any point 
in the perpendicular bisector of we have 
and it is not impossible to have 
, 
. Thus if the circle with centre and radius intersects the circle with centre and radius , then the points of intersection of with satisfy the condition of congruence, then the points of intersection of with 
do not. Thus no more than half of the 8 points of intersection of these circles can satisfy the condition of congruence, i.e. 
.
we have the following example where 
is a regular hexagon, and and 
are the intersections of 
with respectively 
and 
.![[asy] pair A; pair B; pair C; pair D; pair X1; pair X2; pair X3; pair X4;
D=(1,0); C=(-1,0); B=(0.5,sqrt(3)/2);
A=B+C;
X2=-B; X3=-A; X4=X3+D; X1=X2+C;
draw(A--X1--X4--B--A);
draw(X2--C--D--X3);
draw(circumcircle(A,B,C));
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,W);
label("$D$",D,E);
label("$X_1$",X1,S);
label("$X_2$",X2,S);
label("$X_3$",X3,S);
label("$X_4$",X4,S);
[/asy]](http://latex.artofproblemsolving.com/5/f/3/5f3285c775d96676755d625f4a1337b52a555d1e.png)
