be reals such that 
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 
,
and 
.
Y by Adventure10, Mango247
Let 
be an acute angled triangle with 
, 
be the center of its incircle, 
touch points of the incircle with the sides 
and 
respectively, 
the midpoint of the arc 
of the circumcircle of the triangle . 
is the projection of 
on 
is the projection of 
on 
.
a) Prove that that the lines
and 
are parallel.
b) If
is the second point of intersection between the circle of diameter 
and the circumcircle of the triangle , prove that the lines 
and 
intersect at .
be an acute angled triangle with 
, 
be the center of its incircle, 
touch points of the incircle with the sides 
and 
respectively, 
the midpoint of the arc 
of the circumcircle of the triangle . 
is the projection of 
on 
is the projection of 
on 
.a) Prove that that the lines
and 
are parallel.b) If
is the second point of intersection between the circle of diameter 
and the circumcircle of the triangle , prove that the lines 
and 
intersect at .
be the midpoint of minor arc 
.
and 
are both perpendicular to 
,
are collinear and 
lies on the circle with diameter 
)
.
,
are collinear.
.
and 
are both perpendicular to 
,
,
.
