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 
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 son7
Let 
be a right triangle with right angle at the vertex 
. A circle 
with diameter 
intersects the side 
at the point 
, and the tangent to the circle at the point intersects the side 
at point 
. The circle circumscribed around triangle 
intersects the side at points and 
. Determine the ratio of the areas of triangles and 
.
be a right triangle with right angle at the vertex 
. A circle 
with diameter 
intersects the side 
at the point 
, and the tangent to the circle at the point intersects the side 
at point 
. The circle circumscribed around triangle 
intersects the side at points and 
. Determine the ratio of the areas of triangles and 
.This post has been edited 1 time. Last edited by parmenides51, Mar 24, 2024, 11:29 PM
is twice the area of 
which is again twice the area of 
.
stands for small arc 
be the midpoint of 
,
is 
because 
,
are tangents of 
from 
,
is perpendicular bisector of 
,
,
(tangent - chord angle)
Triangles 
![$[ABC]/[BEF] = 2^2=4$](http://latex.artofproblemsolving.com/0/1/a/01a2276ed4ac900e86e1c58dbe10fac515e249de.png)
