[/quote]
,
,
,
and 
lie on a circle 
such that 
.
at 
and 
.
intersect again at 
.
.
and 
.
Equality holds when 
or 
Equality holds when 
intersect its circumcircle 
at 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and that 
and 
.
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
be reals such that 
Prove that 
Let 
Prove that 
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 
,
satisfies 
.
,
.
Y by Adventure10, Mango247
I will quote exactly at every letter.
Let
be a non-isosceles triangle with the incenter 
. Let 
, be the smaller circle through tangent to both 
and 
(summation of indeces is done modulus 3). Let 
, be the other intersection point of 
and 
. Prove that the circumcenters of the triangle 
, 
, 
are collinear.
Let
be a non-isosceles triangle with the incenter 
. Let 
, be the smaller circle through tangent to both 
and 
(summation of indeces is done modulus 3). Let 
, be the other intersection point of 
and 
. Prove that the circumcenters of the triangle 
, 
, 
are collinear.
