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
The triangle 
is acute. The points 
and 
are the feet of altitudes of the triangle from vertices 
and 
, respectively. Given points 
and 
such that and are the internal points of the segments 
and 
, respectively and the angle 
is right. Point 
is the foot of the altitude of the triangle 
drawn from the vertex 
. Prove that the angle 
is right.
is acute. The points 
and 
are the feet of altitudes of the triangle from vertices 
and 
, respectively. Given points 
and 
such that and are the internal points of the segments 
and 
, respectively and the angle 
is right. Point 
is the foot of the altitude of the triangle 
drawn from the vertex 
. Prove that the angle 
is right.
.
and
.
.
still holds.
and let 
and that 
