[/quote]
,
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
matrix 
whose entries are all in 
,
is the matrix of all ones?
and 
Prove that
where 
be a function defined on 
taking values in 
such that
for all 
for all 
;
identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables 
a) of degree 
;
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
.
)
such that there exist integers 
with ![\[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]](http://latex.artofproblemsolving.com/e/f/a/efaa98e77ac0273284ee3446e4335d8cc4f681cb.png)
and 
doubloons, where 
.
coins. Find the number of lucky numbers.
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
be a quadrilateral inscribed in a circle 
such that 
.
and 
be the midpoints of 
and 
respectively. The line 
meets 
.
are concurrent.
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 
.
.
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
