AoPS Academy
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 
. Denote by 
, 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
+1 w
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
, 
and 
be the altitudes of the acute triangle . Prove that the feet of the perpendiculars drawn from point 
on the lines 
and lie on one line.
, and be the altitudes of the acute triangle . Prove that the feet of the perpendiculars drawn from point on the lines and lie on one line.
be orthogonal projections of 
then 
and 
.
are Simson lines of 
or