[/quote]
,
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
be the circumcircle of acute triangle 
.
intersect at 
,
and 
intersect at 
.
,
are on 
and 
such that 
and 
.
are concyclic.
the centre of the circle passing through 
,
are difined similarly. Prove that 
,
,
are concurrent.
be positive real numbers .
> 
that satisfy 
,
€ 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
such that 
.
and 
.
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
?
and 
is a subset of 
such that if 
then 
Find the smallest of 
that satisfy the following property: if 
,
,
,
are all the positive divisors of 
,
divides 
for every 
.
,
:
,
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
