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 circumcircle of the acute triangle 
. The angle bisector of angle intersects 
in the point 
and the short arc of in the point 
. The line through perpendicular to 
intersects the short arc of in 
. The line through 
perpendicular to 
intersects in 
. Prove that 
and lie on a line.
be the circumcircle of the acute triangle 
. The angle bisector of angle intersects 
in the point 
and the short arc of in the point 
. The line through perpendicular to 
intersects the short arc of in 
. The line through 
perpendicular to 
intersects in 
. Prove that 
and lie on a line.This post has been edited 2 times. Last edited by parmenides51, Jan 10, 2020, 1:58 PM
