such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
of positive real numbers such that
,
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 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
Y by Adventure10, Mango247
Squares 
, 
, 
are outwardly drawn on sides of triangle 
. If 
, 
, 
are parallelograms then prove that:
(i) Lines
and 
are orthogonal.
(ii)Triangles and 
have common centroid
, 
, 
are outwardly drawn on sides of triangle 
. If 
, 
, 
are parallelograms then prove that:(i) Lines
and 
are orthogonal.(ii)Triangles
and 
have common centroid
?
, 
and 
are defined by parallelograms mentioned above. 
where 
and 
are the coordinates of two points of one line, and 
and 
are the coordinates of two points of the other line.
be the origin, and 
,
.
.
and 
.
,
.
and 
and the centroid of 
is 
and the centroid of 
is 
as well...
.
, which implies part1. For part2, notice that ![\[ \sum_{cyc}A'-A = i \left(\sum_{cyc}B-C\right)=0\]](http://latex.artofproblemsolving.com/4/5/8/45859c5e5ac4f9836bb9a12faed8f5a246337404.png)
So the two sets of points have the same vector sum and therefore equal centroid.
