[/quote]
,
be a triangle with incenter 
and circumcircle 
.
passing through 
and 
passing through 
are tangent at 
of 
and 
,
of 
and 
.
and 
meet at 
.
be a point such that 
is tangent to 
is tangent to 
are collinear.
.
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
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 
be non-negative real numbers such that 
Prove that
and 
Find the minimum 
.
be a subset of 
.
.
Y by
Fencing Ferdinand wants to fence three rectangular areas. there are fences in three types, with 
amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?
amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?This post has been edited 3 times. Last edited by parmenides51, Dec 14, 2020, 12:55 PM
Reason: removed original wording
Reason: removed original wording
)
or 
)
