[/quote]
,
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
This easy via the binomial theorem for the quantity is just 
,
and 
.
and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
,
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
and let 
be the foot of the
-
and 
respectively. A line through 
and 
and 
and 
are tangent to each other.
with circumcircle 
.
is the angle bisector of 
.
intersects 
.
angle bisector of 
if and only if 
.
be real numbers satisfying
.
,
past 
.
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 
intersect 
.
with orthocenter 
intersects 
at 
intersect 
.
intersects 
and 
intersect on the circumcircle of triangle 
.
Y by
Points 
and 
are the midpoints of sides 
and 
of an acute-angled triangle 
, point 
is the midpoint of segment 
. Point 
is the foot of the altitude drawn from vertex 
to side 
. Through point , circles are drawn tangent to sides and , respectively, at points and . Denote the second intersection point of the circles by 
. Prove that points 
and lie on one straight line.
and 
are the midpoints of sides 
and 
of an acute-angled triangle 
, point 
is the midpoint of segment 
. Point 
is the foot of the altitude drawn from vertex 
to side 
. Through point , circles are drawn tangent to sides and , respectively, at points and . Denote the second intersection point of the circles by 
. Prove that points 
and lie on one straight line.
