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
Points 
and 
are marked respectively on the sides 
abd 
of triangle 
so that 
is the altitude and 
is the bisectrix . It is known that the gravity center of lies on the line 
.
a) Find if =a nad 
=c.
b) Determine all possible values of
for all triangles satisfying the given condition.
and 
are marked respectively on the sides 
abd 
of triangle 
so that 
is the altitude and 
is the bisectrix . It is known that the gravity center of lies on the line 
.a) Find
if =a nad 
=c.b) Determine all possible values of
for all triangles satisfying the given condition.
.
![$\frac{c}{a}\in\left(\frac {3-\sqrt 5}{2}, 1 \right]$](http://latex.artofproblemsolving.com/7/d/1/7d10a6094cd6e671ac198213eb91c10a08587179.png)
and 
with transversal 
we get: 
