[/quote]
,
and 
on the plane such that 
is an equilateral . Let 
and 
at 
and 
respectively and 
intersect 
and 
at 
respectively .
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
.
on 
and 
on 
such that 
Interior points.
)
.
. 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
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
The square 
is inscribed in the right triangle 
(with 
) so that points 
, 
lie on the legs 
and 
respectively ,and points 
, 
lie on the hypotenuse 
. The circumcircle of triangles 
an 
intersect at and 
. Prove that the lines 
and 
meet on the hypotenuse .
is inscribed in the right triangle 
(with 
) so that points 
, 
lie on the legs 
and 
respectively ,and points 
, 
lie on the hypotenuse 
. The circumcircle of triangles 
an 
intersect at and 
. Prove that the lines 
and 
meet on the hypotenuse .This post has been edited 1 time. Last edited by NikitosKh, Mar 31, 2019, 3:18 PM
Reason: misprint
Reason: misprint
so 
and 
and 
with 
