[/quote]
,
be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side 
,
the midpoint of 
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, 
)
are there such that 
and 
?
be an acute triangle with 
,
onto 
.
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](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 
.
.
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 
,
,
is a positive integer.
of positive integers 
such that 
.
denotes component-wise function multiplication)
and 
Prove that
distinct points 
connected if the line 
points from 
such that there exists a set 
is connected with exactly 
other points from 
, where 
is a real number. Prove that for all positive 
then prove
Y by
The circles 
and 
intersect at points 
and 
, are inside the circle 
and tangent to it at points 
and , respectively. The line 
intersects the circle at points 
and 
. Let 
and 
be the second intersection points of the lines 
and 
with . Let and 
be the second intersection points of 
and 
with . Prove that the points 
and 
lie on the same circle.
and 
intersect at points 
and 
, are inside the circle 
and tangent to it at points 
and , respectively. The line 
intersects the circle at points 
and 
. Let 
and 
be the second intersection points of the lines 
and 
with . Let and 
be the second intersection points of 
and 
with . Prove that the points 
and 
lie on the same circle.
a
is the radical axis o
a
a
i
i
