[/quote]
,
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 
.
,
the midpoints of 
,
.
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 
other than 
.
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its circumradius. Let 
,
across the line 
,
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
is inscribed in circle 
.
its 
.
meets 
at 
and 
meets 
a second time at 
meets 
a second time at 
.
passes through the antipodal to 
Y by
In the convex quadrilateral 
on the side 
the points 
and 
are selected (the point lies between 
and ), and on the side 
the points 
and 
are selected (point lies between 
and 
). It turned out that 
and 
. You can also circumscribe a circle around the quadrilateral 
. Prove that a circle can also be circumscribed around the quadrilateral 
.
on the side 
the points 
and 
are selected (the point lies between 
and ), and on the side 
the points 
and 
are selected (point lies between 
and 
). It turned out that 
and 
. You can also circumscribe a circle around the quadrilateral 
. Prove that a circle can also be circumscribed around the quadrilateral 
.This post has been edited 4 times. Last edited by parmenides51, Jul 13, 2021, 9:57 PM
