be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
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)
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 
.
Y by Adventure10, Mango247
Let 
be the the isogonal conjugate of 
with respect to triangle 
, and 
are in the interior of triangle . Denote by 
the circumcenters of triangle 
, 
the circumcenters of triangle 
, 
the circumcenter of triangle 
, 
the circumcenter of triangle 
. Prove that 
is parallel to 
.
be the the isogonal conjugate of 
with respect to triangle 
, and 
are in the interior of triangle . Denote by 
the circumcenters of triangle 
, 
the circumcenters of triangle 
, 
the circumcenter of triangle 
, 
the circumcenter of triangle 
. Prove that 
is parallel to 
.
be the circumcenter of the 
Let 
be the pedal triangles of 
;
of 
is their common circumcenter. The midpoint 
of 
is the circumcenter of the 
on account of the rigth angles 
The isogonal 
of 
with respect to the angle 
is its A-altitude 
But 
as well 
and similarly, 
The 
are centrally similar, having the correspnding sides parallel. 
are their corresponding circumcenters and 
are their corresponding points, the concurrency points of their corresponding cevians 
Their corresponding lines 
are parallel. In ezactly the same way, 
be the circumcenter of 
.
.
.
is the perpendicular bisector of 
,
.
are concyclic
are concyclic and 
are concyclic.
,
are collinear at the radical axis of 
and 
. i.e. 
is perpendicular to 
... 
is the circumcenter of 
. i.e. 
lie on the perpendicular bisector of 
.
,
.
are coaxial
be the isogonal conjugate of 
.
be the circumcenter of 
.
be the circumcenter of 
.
be the circumcenter of 
.
