[/quote]
,
and 
of an equilateral triangle 
are taken points 
and 
,
, points 
and 
so that 
.
and 
is equal to 
.
. Prove that: 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
over 
.
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
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)
Y by
A square with a side of 
cm was drawn on the board.Alexey crossed it with two straight lines parallel to one pair of sides of the square. After that Danil crossed the square two straight lines parallel to another pair of sides of the square. As a result, the square broke into 
rectangles, and it turned out that the lengths of the sides of the central section are 
cm and 
cm. Find the sum of the areas of the rectangles at corners.
cm was drawn on the board.Alexey crossed it with two straight lines parallel to one pair of sides of the square. After that Danil crossed the square two straight lines parallel to another pair of sides of the square. As a result, the square broke into 
rectangles, and it turned out that the lengths of the sides of the central section are 
cm and 
cm. Find the sum of the areas of the rectangles at corners.
