[/quote]
,
.
and ray 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
and 
on the plane such that 
is an equilateral . Let 
and 
at 
and 
at 
,
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 
Y by
In the isosceles triangle 
, let 
and 
be points on the sides 
and 
, respectively, such that the lines 
and 
intersect at 
. Furthermore, the triangles 
and 
are equal and have 
and 
. Calculate ![$\frac{[ADFE]}{[ABC]}$](//latex.artofproblemsolving.com/5/c/a/5caef8d812622fae7af8919c479205df3aa3badf.png)
.
Clarification:![$[ABC]$](//latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
denotes the area of the figure .
, let 
and 
be points on the sides 
and 
, respectively, such that the lines 
and 
intersect at 
. Furthermore, the triangles 
and 
are equal and have 
and 
. Calculate ![$\frac{[ADFE]}{[ABC]}$](http://latex.artofproblemsolving.com/5/c/a/5caef8d812622fae7af8919c479205df3aa3badf.png)
.Clarification:
![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
denotes the area of the figure .
.
.
.![$\frac{[ADFE]}{[ABC]}=\frac{1}{6}$](http://latex.artofproblemsolving.com/b/b/d/bbdb030b6a0770172d4a1f64803f56432365b1f0.png)
.
