[/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 
.
,
.
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 
in line 
and 
respectively. A line through 
and 
respectively such that 
while 
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 Adventure10, Mango247
Let 
and 
be the centers of the circumscribed and inscribed circle the acute-angled triangle 
, respectively. It is known that line 
is parallel to the side 
of this triangle. Line 
, where 
is the midpoint of , intersects the altitude 
at the point 
. Find the length of the segment 
, if the radius of the circle inscribed in the triangle is equal to 
.
(Grigory Filippovsky)
and 
be the centers of the circumscribed and inscribed circle the acute-angled triangle 
, respectively. It is known that line 
is parallel to the side 
of this triangle. Line 
, where 
is the midpoint of , intersects the altitude 
at the point 
. Find the length of the segment 
, if the radius of the circle inscribed in the triangle is equal to 
.(Grigory Filippovsky)
This post has been edited 2 times. Last edited by parmenides51, Oct 11, 2020, 10:22 PM
