[/quote]
,
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 
.
be a scalene triangle with orthocenter 
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 incenter 
and let 
,
are selected on sides 
,
such that 
,
,
.
intersects 
.
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its circumradius. Let 
,
across the line 
,
.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
is a prime number
Y by Mango247
Points 
and 
on side
of convex quadrilateral 
are such that 
. Point 
is a non-
intersection point of the circumscribed circles of triangles 
and 
, and point 
is a non-
intersection point of the circumscribed circles of triangles 
and 
. Prove that points 
and lie on the same circle.
and 
on side
of convex quadrilateral 
are such that 
. Point 
is a non-
intersection point of the circumscribed circles of triangles 
and 
, and point 
is a non-
intersection point of the circumscribed circles of triangles 
and 
. Prove that points 
and lie on the same circle.
,
and 
,
and 
,
