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 
.
be a scalene triangle with orthocenter 
and circumcenter 
.
,
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
Y by
The diagonals of the cyclic quadrilateral 
intersect at the point 
. Let 
and 
are the centers of the circles circumscribed around the triangles 
and 
, respectively. A straight line passing through the point parallel to 
, and a straight line passing through the point parallel to 
, intersect at the point 
. Prove that the point lies on segment 
.
intersect at the point 
. Let 
and 
are the centers of the circles circumscribed around the triangles 
and 
, respectively. A straight line passing through the point parallel to 
, and a straight line passing through the point parallel to 
, intersect at the point 
. Prove that the point lies on segment 
.This post has been edited 1 time. Last edited by parmenides51, Apr 29, 2021, 8:08 PM
