[/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 
.
and 
.
is chosen on the segment 
.
and 
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 
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
Point 
is the center of circumcircle 
of the isosceles triangle 
(
). Bisector of the angle 
intersects at the point 
. Point 
is the center of the circumcircle of the triangle 
. Construct the triangle given the points 
.
(Andrey Mostovy)
is the center of circumcircle 
of the isosceles triangle 
(
). Bisector of the angle 
intersects at the point 
. Point 
is the center of the circumcircle of the triangle 
. Construct the triangle given the points 
.(Andrey Mostovy)
This post has been edited 3 times. Last edited by parmenides51, Oct 11, 2020, 10:19 PM
