blocks, each weighing at least 
;
.
with 
you can choose a subset of the blocks whose total weight is at least 
.
such that 
.
chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes 
rows and 
cells in the intersection. Babek wants to have all these 
be non-negative real numbers such that 
Prove that
be a circle, and let a line 
intersect 
and 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
.
.
and 
be points on a plane such that 
,
be the set of all points 
,
is a real number. The path that 
Y by
In a sphere of radius 
, consider a maximum circle 
. On one of the hemispheres determined by consider three circles of equal radii, tangent to each other, in pairs, and tangent to . Determine the radius of another circle, located on the same hemisphere and that is tangent to the three circles.
, consider a maximum circle 
. On one of the hemispheres determined by consider three circles of equal radii, tangent to each other, in pairs, and tangent to . Determine the radius of another circle, located on the same hemisphere and that is tangent to the three circles.This post has been edited 1 time. Last edited by parmenides51, Oct 18, 2021, 6:53 PM
