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
intersect 
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 
.
.
.
,
be the set of all points 
,
is a real number. The path that 
Y by Adventure10
In the acute-angled triangle 
with circumcircle 
and orthocenter 
, points 
and 
are the feet of the perpendiculars from 
onto 
and from 
onto 
respecively. Let 
be a point on the minor arc of . Points 
and 
are the feet of the perpendiculars from onto lines and respectively. Let 
and 
intersect at 
. Prove that 
.
with circumcircle 
and orthocenter 
, points 
and 
are the feet of the perpendiculars from 
onto 
and from 
onto 
respecively. Let 
be a point on the minor arc of . Points 
and 
are the feet of the perpendiculars from onto lines and respectively. Let 
and 
intersect at 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jul 13, 2019, 8:26 AM
Reason: added sentence ''Let PH and MN intersect at R.''
Reason: added sentence ''Let PH and MN intersect at R.''
s
at 
.
,
.
.
and 
is isosceles. 
.![\[\angle RMP = 90 + \angle XMD = 90 + (90-\angle MXD) = \angle RXH\]](http://latex.artofproblemsolving.com/9/e/9/9e97dbf85c0fe3158d33f733090993c683e2fd38.png)
.
,
so 
and we're done. 
from 
