[/quote]
,
be given positive integer. Suppose that there is only one positive integer 
which satisfies the equation
for some positive integers 
.
be positive real numbers such that 
.
knowing that the denominators are positive real numbers.
such that 
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
and 
as the orthocenter and incenter, respectively, of triangle 
.
be the midpoint of 
.
if and only if 
.
be an acute, scalene triangle with circumcenter 
and symmedian point 
.
be the point on the circumcircle of triangle 
such that 
.
.
,
and 
,
is the perpendicular bisector of 
.
and 
,
,
.
at 
.
,
.
and 
are tangent to each other
prove that
such that
are being replaced with 
.
will appear on the board.
.
lie on segments 
respectively. Points 
are such that 
and 
are equlateral, 
lie on opposite sides of 
,
.
.
be the circumcircle of a triangle 
the midpoints of the sides 
and 
,
the midpoint of the arc 
and 
intersect the perpendicular bisectors of 
,
and 
intersect at 
.
,
intersects 
at 
,
intersects 
at 
such that: 
=
.
,
,
on circle 
is a isosceles triangle
Y by
Given a set 
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following:
- and are non-empty subsets from ,
- the sum of all elements in each of and are equal, and
-
is an empty subset.
with exactly 9 elements that is subset of 
. Prove that there exist two subsets 
and 
that satisfy the following:-
and are non-empty subsets from ,- the sum of all elements in each of
and are equal, and-
is an empty subset.
(although this is a very common problem idea, perhaps this exact problem already exists somewhere)
.
be a set with exactly 
elements. Prove that there exist two subsets 
,
for which 
.![\[\binom{9}{1}+\binom{9}{2}+\dots+\binom{9}{6}= 465\]](http://latex.artofproblemsolving.com/3/2/a/32a7ca3d34895c1ebeb9b63231d599e3f6e56582.png)
sets that satisfy. Observe that the sum of all elements in 
.
for which 
where the sum of all elements of 
sets 
respectively. Since equality is reached, the set 
.
.
has the same sum of elements as 
,
for which the sum of all elements of 
and 
,
satisfy all three criteria 
