[/quote]
,
be an acute-angled triangle, and let 
and 
be two points on its side 
.
in such a way that the convex quadrilateral 
is cyclic, 
,
.
in such a way that the convex quadrilateral 
is cyclic, 
,
.
by 
and 
such that 
.
square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by 
.
with 
and 
are of the same color.
points such that these 
where 
is a positive real number instead of 
such that if 
,
by 
also belongs to 
.
be a random point on the smaller arc 
,
be the intersection point of segments 
.
to the circumcircle of the triangle 
are 
is the center of the square. Prove that points 
,
and 
.
a regular hexagon inscribed in a circle 
.
be a point inside 
its polar reflection wrt 
meet 
.
the polygon formed by the vertices 
using 
,
,
,
,
intersects lines 
and 
at 
and 
,
and 
be distinct points on side 
.
be the circumcircle of 
.
.
,
.
satisfies the quadratic equation 
and deduce that the value of 
.
.
,
is 
.
.
.
where 
and 
defined as
is given by
denotes the number of points of intersection between the curves
as
Number of subsets of 
whose sum of elements is divisible by 
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
