[/quote]
,
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
the intersection of 
and 
is the reflection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
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 
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 
.
on 
and 
on 
such that 
Interior points.
)
.
,
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
Y by
Let 
be a triangle. Let 
be the foot of altitude from 
on 
, and 
be the foot of altitude from 
on 
. Let 
be the point such that 
is a parallelogram. Denote by 
the second intersection point of the circumscribed circles of the triangles 
and 
. Let 
be the midpoint of and 
be the midpoint of 
. It is known that the points , and are collinear. Calculate the ratio 
be a triangle. Let 
be the foot of altitude from 
on 
, and 
be the foot of altitude from 
on 
. Let 
be the point such that 
is a parallelogram. Denote by 
the second intersection point of the circumscribed circles of the triangles 
and 
. Let 
be the midpoint of and 
be the midpoint of 
. It is known that the points , and are collinear. Calculate the ratio 
This post has been edited 1 time. Last edited by parmenides51, Nov 7, 2021, 7:18 AM
