[/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 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
and let 
be the foot of the
-
.
and 
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 
.![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](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 
.
and 
intersect on the circumcircle of triangle 
.
Y by Adventure10
Let 
be a triangle with circumcenter 
. Point 
are chosen on sides 
and 
, respectively, such that 
is a rhombus. The circumcircles of 
and 
intersect 
at 
and 
respectively. Show that 
.
Proposed by Ariel García
be a triangle with circumcenter 
. Point 
are chosen on sides 
and 
, respectively, such that 
is a rhombus. The circumcircles of 
and 
intersect 
at 
and 
respectively. Show that 
.Proposed by Ariel García
This post has been edited 1 time. Last edited by plagueis, May 27, 2020, 2:41 PM
.
(pop) which is easy by congruent triangles with some angles.
intersect the circumcircle again at 
,
intersect the circumcircle again at 
.
is an isosceles trapezoid with 
,
and 
,
