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 
.
.
.
,
be the set of all points 
,
is a real number. The path that 
Y by
Given an isosceles triangle 
with vertex 
, circle 
. Tangents to this circle at the points 
and 
intersect at the point 
. Consider the circle 
, located inside 
, which touches the circle 
and the side 
at the point 
. Circle 
is also inside and touches circle 
and circle at point 
, and the sides 
at the point 
. Prove that the bisector of 
passes through the incenter of 
.
with vertex 
, circle 
. Tangents to this circle at the points 
and 
intersect at the point 
. Consider the circle 
, located inside 
, which touches the circle 
and the side 
at the point 
. Circle 
is also inside and touches circle 
and circle at point 
, and the sides 
at the point 
. Prove that the bisector of 
passes through the incenter of 
.
