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 a point in the plane of a triangle 
, not lying on the lines 
, or 
. Denote by 
the intersections of the parallels through 
to the lines 
with the line 
. Define analogously 
. Prove that 
lie on the same conic.
Proposed by Mihai Miculita, Oradea, Romania
be a point in the plane of a triangle 
, not lying on the lines 
, or 
. Denote by 
the intersections of the parallels through 
to the lines 
with the line 
. Define analogously 
. Prove that 
lie on the same conic.Proposed by Mihai Miculita, Oradea, Romania
.
,
,
meet 
,
,
,
,
respectively.
to these four pairs of triangles we obtain four collinearities:
to the hexagon 
