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
Let ABC be a triangle with angle B unequal to angle C. The incircle I of ABC touches the sides BC, CA, AB at the points
D, E, F respectively. Let AD intersect I at D and P. Let Q be the intersection of the lines EF and the line passing through P and perpendicular to AD, and let X, Y be intersections of the line AQ and DE, DF, respectively. Show that the point A is the midpoint of XY .
D, E, F respectively. Let AD intersect I at D and P. Let Q be the intersection of the lines EF and the line passing through P and perpendicular to AD, and let X, Y be intersections of the line AQ and DE, DF, respectively. Show that the point A is the midpoint of XY .
and let 
be the diametral opposite point of 
wrt. circle 
.
collinear.Since the quadrilater 
is harmonic we obtain 
where 
is the midpoint of side 
so we get that lines 
and 
.
.
.
but 
is the midpoint of segment 
.
