[/quote]
,
such that:
for any positive integer 
,
are all distinct, and
,
?
and let 
be positive integers such that 
.
and determine when equality holds.
different ropes. She then performs a experiment that causes 
children, each connected to each other with the same type of rope, to be cured. Two experiments are said to be of the same type, if each of the ropes connecting the children has the same medicine imbued in it. She then unties them and lets them go back home.
be the minimum m such that Myla can perform at least 
For every 
there exists a 
and 
such that for all 
,![\[f(n, k) = a_kn + b_k.\]](http://latex.artofproblemsolving.com/6/6/0/660699c6cec811eee25f727076665a6df3257f57.png)
Find the value of 
for every 
inscribed in a circle with center 
.
is any point not on (O). 
intersect 
at 
.
be the Lemoine points of triangle 
respectively. Prove that 
are collinear.
be a sequence of positive real numbers such that for every 
,![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
,![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
and 
lie on 
and 
respectively such that 
.
and 
are the reflections of 
such that the circumcenter of triangle 
coincides with the circumcenter of triangle 
.
such that : 
such that : 
and 
such that 
.
written on the first page and 
be the largest possible number such that for any 
,
such that for all 
,![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
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
