[/quote]
,
be a triangle, and let 
be a point on side 
.
at 
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
Let 
be an inscribed quadrilateral. Let the incenters of 
and 
be 
and 
respectively. Let the intersection point of the line that passes through and perpendicular to 
and the line that passes through and perpendicular to 
be 
. Prove that 
be an inscribed quadrilateral. Let the incenters of 
and 
be 
and 
respectively. Let the intersection point of the line that passes through and perpendicular to 
and the line that passes through and perpendicular to 
be 
. Prove that 
is the midpoint of 
.
,
is a kite. 
and 
intersect at the midpoint 
of the arc 
(which does not contain vertex 
,
.
,
.
.
and
we deduce that 
,
.
