AoPS Academy
be positive integers and let 
be a non-increasing list of positive real numbers. Prove that there exists 
sets 
which partition the set 
such that 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
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 
. Prove that the points 
and lie on the same line.
, 
, and 
, are marked on the plane. You have a pencil and a double-edged ruler of width 
. Using them, you may perform the following operations:
and 
are marked, draw the line connecting and .
and 
are drawn, mark the intersection of and .
is drawn, draw a line parallel to that is at distance away from (note that two such lines may be drawn).
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)
be an equilateral triangle and let 
be an inner point of the side 
. A circle is tangent to at and intersects the sides 
and 
in the inner points 
and 
respectively. Prove that 
.
be an equilateral triangle and let be an inner point of the side . A circle is tangent to at and intersects the sides and in the inner points and respectively. Prove that .
