[/quote]
,
be a set of Natural numbers from 
to 
)
.
and 
,
is not less than the area of 
,
inequalities, these given comparisons satisfy the following three constraints:
and 
,
and 
respectively, then the sum of elements of the union 
is greater than that of 
.
that satisfies all these conditions?
.
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the orthocenter of 
be the point of intersection of the circumcircle 
of the triangle 
,
be the point of intersection (other than 
)
and the circle 
must hold.
the length of the line segment 
such that 
(where 
)
that satisfies 
for all 
is the function obtained by composing 
times, that is, 
)
be an acute triangle with 
,
be the projection from 
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
intersect 
.
.
sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
,
,
of positive integers 
such that 
.
denotes component-wise function multiplication)
and 
Prove that
Y by
Let 
be colinear points in this order, 
an arbitrary circle passing through 
and 
, and 
an arbitrary line different from 
, passing through A and intersecting at 
and 
. The bisectors of the angles 
and 
intersect at 
and 
. Prove that 
.
be colinear points in this order, 
an arbitrary circle passing through 
and 
, and 
an arbitrary line different from 
, passing through A and intersecting at 
and 
. The bisectors of the angles 
and 
intersect at 
and 
. Prove that 
.
lies in this order
are concyclic 
