[/quote]
,
and 
then prove 
and 
.
are positive real numbers such that 
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
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 
.
and circumcenter 
.
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and let 
,
such that 
,
,
.
intersects 
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
Y by
Let 
be a triangle with a circumcircle 
. The line perpendicular on 
through 
intersects and at 
and 
, respectively. Let 
and 
be the feet of the altitudes from on 
and 
, respectively. The perpendicular on 
from cuts at 
. Let 
and 
be midpoints of the segments 
, 
, and 
. Show that 
, 
, and 
are collinear.
Alternative version:
Let be a triangle with circumcircle and circumcenter 
. The line perpendicular on from intersects and at and , respectively. Let and be feet of the altitudes from to and , respectively. Let and be midpoints of the segments and . Show that is circumcenter of 
.
be a triangle with a circumcircle 
. The line perpendicular on 
through 
intersects and at 
and 
, respectively. Let 
and 
be the feet of the altitudes from on 
and 
, respectively. The perpendicular on 
from cuts at 
. Let 
and 
be midpoints of the segments 
, 
, and 
. Show that 
, 
, and 
are collinear.Alternative version:
Let
be a triangle with circumcircle and circumcenter 
. The line perpendicular on from intersects and at and , respectively. Let and be feet of the altitudes from to and , respectively. Let and be midpoints of the segments and . Show that is circumcenter of 
.
