[/quote]
,
such that 
for all 
,
.
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 Adventure10, Mango247
At the exterior of the square 
it is constructed the isosceles triangle 
with 
is the intersection of the bisector line of the angle 
with its perpendicular that passes through 
is the intersection of the 
with its perpendicular that passe through 
is the intersection of 
with 
If
is the center of gravity of the triangle 
prove that 
and 
are parallel.
it is constructed the isosceles triangle 
with 
is the intersection of the bisector line of the angle 
with its perpendicular that passes through 
is the intersection of the 
with its perpendicular that passe through 
is the intersection of 
with 
If
is the center of gravity of the triangle 
prove that 
and 
are parallel.
and label the points as follows
Dropping an altitude from 
.
and 
,
and 
.
and using algebra, we see that 
.
and to finish off we see that the slope of 
and 
are both 
,
