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 Adventure10, Mango247
On the plane is given the acute triangle 
. Let 
and 
be the feet of the altitudes of 
and 
drawn from those vertices, respectively. Tangents at points and drawn to the circumscribed circle of the triangle 
intersect at 
. Prove that the circles circumscribed around the triangles 
, 
and have a common point.
. Let 
and 
be the feet of the altitudes of 
and 
drawn from those vertices, respectively. Tangents at points and drawn to the circumscribed circle of the triangle 
intersect at 
. Prove that the circles circumscribed around the triangles 
, 
and have a common point.This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2022, 2:47 AM
