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 center of the circumcircle of the acute-angled triangle 
. Let 
and 
be the circumcenters of triangles 
and 
respecrively. For which triangles is the triangle 
similar to the starting triangle (with a suitable order of the vertices)?
be the center of the circumcircle of the acute-angled triangle 
. Let 
and 
be the circumcenters of triangles 
and 
respecrively. For which triangles is the triangle 
similar to the starting triangle (with a suitable order of the vertices)?This post has been edited 2 times. Last edited by parmenides51, Jan 11, 2020, 7:55 PM
