[/quote]
,
.
as 
,
and 
as 
.
.
be a set of Natural numbers from 
to 
.
)
.
and 
,
,
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, 
)
are there such that 
and 
?
,
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 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
, where 
is a real number. Prove that for all positive 
then prove
Y by
In a triangle 
, 
is the incenter, and 
is the center of the tangent circle for the side 
and the extensions of the other sides. A point 
is chosen on the arc 
of the circumcircle of 
, so that 
is a half of 
. 
is the midpoint of the arc of the circumcircle of 
. Prove that the points 
lie on one straight line.
(O. Pyaive)
, 
is the incenter, and 
is the center of the tangent circle for the side 
and the extensions of the other sides. A point 
is chosen on the arc 
of the circumcircle of 
, so that 
is a half of 
. 
is the midpoint of the arc of the circumcircle of 
. Prove that the points 
lie on one straight line.(O. Pyaive)
