[/quote]
,
has an incenter 
Suppose 
.
.
meets 
again at point 
.
and 
.
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
at 
and 
.
.
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
.
where 
and 
again at 
or 
concur on 
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
,
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say 
and erase the number in the other square.
,
and circumcentre 
.
and 
lie on 
and 
.
meet 
meet 
and 
have an intersection lie on line 
.
be a convex hexagon with 
and 
.
and 
.
and 
are the centers of mass of 
and 
,
.
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
a variable point on segment 
.
is constructed such that 
and 
.
and 
.
,
,
and that 
,
.
.
inmixtilinear circle is tangent to 
at 
respectively. 
and 
and 
and 
,
be the projection from 
.
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 
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
and 
intersect 
and 
at 
.
,
and that line 
is not parallel to line 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
are perpendicular.
and 
are perpendicular.
.
and 
,
is not less than the area of 
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 
Y by
Let 
be a scalene acute triangle, where 
is the internal bisector of 
and 
is the midpoint of 
. Let the internal bisectors of 
and 
intersect 
and 
in 
and 
, respectively. Prove that the circumcircle of 
is tangent to if and only if 
belongs to it.
be a scalene acute triangle, where 
is the internal bisector of 
and 
is the midpoint of 
. Let the internal bisectors of 
and 
intersect 
and 
in 
and 
, respectively. Prove that the circumcircle of 
is tangent to if and only if 
belongs to it.
so 
so 
then 
passes through midpoint of arc 
meet 
would be cyclic so 
