be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
and 
Prove that
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
Y by Adventure10, Mango247
Let 
be an acute triangle such that 
. Let 
be points such that 
are tangent to the circumcircle of and 
(
is on one side of line 
and are on the other side). Let 
be intersections of line 
and lines 
. Let 
be intersection of 
and 
, and 
be intersection of 
and 
. Prove that 
.
be an acute triangle such that 
. Let 
be points such that 
are tangent to the circumcircle of and 
(
is on one side of line 
and are on the other side). Let 
be intersections of line 
and lines 
. Let 
be intersection of 
and 
, and 
be intersection of 
and 
. Prove that 
.
is not equal to 
