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 
.
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.
Y by Adventure10
In a given trapezium 
with 
parallel to 
and 
, the line 
bisects the angle 
. The line through 
parallel to 
meets the segments and 
in 
and 
, respectively. Let 
be the circumcenter of the triangle 
. Suppose that 
. Prove the equality
![\[ CF = AF + FO .\]](//latex.artofproblemsolving.com/2/2/5/22572c6ee88506a3bf12ba777302654282ec4cc8.png)
with 
parallel to 
and 
, the line 
bisects the angle 
. The line through 
parallel to 
meets the segments and 
in 
and 
, respectively. Let 
be the circumcenter of the triangle 
. Suppose that 
. Prove the equality![\[ CF = AF + FO .\]](http://latex.artofproblemsolving.com/2/2/5/22572c6ee88506a3bf12ba777302654282ec4cc8.png)
.
is the circumcentre of 
so 
.
So 
and 
are congruent by SAS 
cyclic. So 
.
is equilateral, so 
,
and 
,
.
,
.
is cyclic, since 
.
is equilateral, since 
combined with 
.
is equilateral, meaning that 
and the problem is solved.
