[/quote]
,
,
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.
.
can be written as the sum of two square numbers
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 
.
for which 
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 projection of 
.
be the intersection of 
and 
be the points on 
,
are on a line perpendicular to 
intersect 
,
intersects the circumcircle of 
.
Y by Adventure10, Mango247
Let 
be a sequnce of concentric circles such that the sequence 
, where 
denotes the radius of 
, is increasing and 
as 
. Let 
be a triangle inscribed in 
. extend the rays 
to meet 
in 
and 
respectively and form the triangle 
. Continue this process. Show that the sequence of triangles 
tends to an equilateral triangle as 
be a sequnce of concentric circles such that the sequence 
, where 
denotes the radius of 
, is increasing and 
as 
. Let 
be a triangle inscribed in 
. extend the rays 
to meet 
in 
and 
respectively and form the triangle 
. Continue this process. Show that the sequence of triangles 
tends to an equilateral triangle as 
