[/quote]
,
be positive integers and let 
be a non-increasing list of positive real numbers. Prove that there exists 
which partition the set 
such 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 
are marked, draw the line connecting 
and 
are drawn, mark the intersection of 
is drawn, draw a line parallel to 
using these operations.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
,
to be the set of all triplets 
of natural numbers, with 
odd and 
,
and 
.
be positive real numbers . Prove that
and 
in the plane, draws the perpendicular bisector of 
.
.
be the power of 
be the power of 
.
satisfy the equality
Find the maximum possible value of 
.
board. Julia
,
be its 
be the intersection of 
and 
be the points on 
,
are on a line perpendicular to 
intersect 
,
intersects the circumcircle of 
.
Y by Tumon2001, Adventure10, and 1 other user
Let 
, 
and 
be three points on a line (in this order).
For each circle
through the points and , let 
be one point of intersection of the perpendicular bisector of 
with the circle . Further, let 
be the second point of intersection of the line 
with .
Show that for each circle, the ratio of lengths 
is the same.
, 
and 
be three points on a line (in this order).For each circle
through the points and , let 
be one point of intersection of the perpendicular bisector of 
with the circle . Further, let 
be the second point of intersection of the line 
with .Show that for each circle
, the ratio of lengths 
is the same.
is tangent to 
and 
(
and from this also 
) So,by similarity of 
, 
,we get 
and 
As 
.Therefore 
Hence the ratio is fixed.
and 
be two circles passing through 
and 
.
= 
and 
.
= 
and 
,
= 
and let 
= 
.
and let 
= 
.
and 
.
is cyclic.
+ 
= 
°
= 
°
is also cyclic.
= 
is a diameter of 
passes through 
= 
(angle bisector theorem) = a constant.
