[/quote]
,
be a positive integer. Consider necklaces of length n whose beads are colored in one of three colors, say red, green, or blue, with exactly two beads of each color (so 
)
of such distinct necklaces (up to rotation and reflection).
and 
meet at two distinct points 
and 
.
through 
(other than 
.
be the midpoint of the minor arc 
(i.e., the point on 
subtends the arc 
be the midpoint of the minor arc 
.![\[
MN \parallel \text{(radical axis of } \omega_1, \omega_2).
\]](http://latex.artofproblemsolving.com/6/6/4/6647f0f7182bdf5096ec5f2956a272803082f4f7.png)
be the foot of the altitude from the vertex 
.
be the midpoint of side 
,
be the circumcentre of 
meets line 
at 
.
,
such that : 
is on 
.
.
is the circumcenter of 
,
is the circumcenter of 
intersects with 
at 
with no obtuse angles.![$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$](http://latex.artofproblemsolving.com/d/1/4/d14154281d7ccf60a2dcfe1d937b974ae1d74f97.png)
be its incenter, 
be its 
-
be midpoint of arc 
,
.
be the intersection of the line 
with 
,
be the intersection of the angle bisector of 
with the line 
,the intersection of line 
with 
and 
,the intersection of 
and 
such that 
.
are collinear. Who has a winning strategy?
be an acute angled triangle with orthocenter 
and 
.
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
and let line 
intersect line 
.
and 
)
intersect the circumcircle of 
again at 
)
concur.
be real numbers satisfying 
Prove that 
and determine all cases of equality.
such that:
for any positive integer 
,
are all distinct, and
,
?
Y by
Let 
and 
be two points that are diametrically opposite to each other on a unit sphere. 
right square pyramids are fitted along the line segment 
, such that the apex and altitude of each pyramid 
, where 
, are 
and 
respectively, and the points 
are collinear.
(a) Find the maximum total volume of pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of .
(b) Find the maximum total volume of pyramids that can be fitted in the sphere, in terms of .
(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as tends to infinity.
Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).
and 
be two points that are diametrically opposite to each other on a unit sphere. 
right square pyramids are fitted along the line segment 
, such that the apex and altitude of each pyramid 
, where 
, are 
and 
respectively, and the points 
are collinear.(a) Find the maximum total volume of
pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of .(b) Find the maximum total volume of
pyramids that can be fitted in the sphere, in terms of .(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as
tends to infinity.Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).
This post has been edited 2 times. Last edited by smartvong, Apr 13, 2025, 5:09 PM
