[/quote]
,
is a prime number
satisfying
Prove
is a positive integer. Call all positive divisors of 
and 
beautiful divisors and difference of any 
that verify the following equation for all 
:
and circumcircle 
.
passing through 
and 
passing through 
are tangent at 
and 
,
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 
cells in the intersection. Babek wants to have all these 
be non-negative real numbers such that 
Prove that
and 
Find the minimum 
Y by Adventure10, Mango247
Triangle 
has circumcentre 
and centroid 
. The lines 
and 
are perpendicular. Let meet the circumcircle of again at 
. Lines 
and 
intersect at 
and 
and 
intersect at 
.Prove that the circumcentre of triangle 
lies on the circumcircle of .
has circumcentre 
and centroid 
. The lines 
and 
are perpendicular. Let meet the circumcircle of again at 
. Lines 
and 
intersect at 
and 
and 
intersect at 
.Prove that the circumcentre of triangle 
lies on the circumcircle of .
are modpoints of 
be the 
enlargement transformation centred on 
.
,
.
,
and 
.
;
,
.
is a mid-segment of 
,
is a mid-segment of 
,
.
.
.
,
be the midpoints of 
,
,
.
is a parallelogram. Hence, 
,
.
,
