[/quote]
,
![$f : [0,1] \longrightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/9/a/4/9a4b7997c35990f839742d29866122791ade8b7c.png)
is differentiable with 
.
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
,
for all 
.
positive real numbers satisfying 
,![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
is satisfied. Point 
is taken on the circumcircle of 
such that 
.
which passes from 
and 
respectively at 
and 
.
passes from circumcenter of 
.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
.
,
point on AC, AB respectively such that 
,
.
,
lines intersect at point 
.
,
is differentiable and 
for all 
.
,
.
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
is called the period of 
.
of period 
.
)
and let 
be positive integers such that 
.
and determine when equality holds.
denote the set of natural numbers, and let 
,
,
.
whose product is a perfect cube.
prove that the triangle must have a right angle.
a prime number, with 
and 
,
vector space.
is a group?
be six positive real numbers. Prove that
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
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
