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
Point 
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.
Prove that
.
Vadzim Kamianetski
is the foot of the altitude from 
of triangle 
. On the lines 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles and at and intersect at 
.Prove that
.Vadzim Kamianetski
This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 6:45 PM
hence 
are concyclic. If 
is the circumcenter of 
,
are collinear, 
and 
is the reflection of 
.
and 
are tangent 
so 
