AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
is a prime number
satisfying
Prove
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
that verify the following equation for all 
:
be a triangle with incenter 
and circumcircle 
. Circles 
passing through 
and 
passing through 
are tangent at . Let meet minor arc 
of at 
and at 
, and let meet minor arc 
of at 
and at 
. Rays 
and 
meet at 
. Let 
be a point such that 
is tangent to and 
is tangent to .
are collinear.
be an acute triangle and let be an interior point for it such that 
. Prove that
When does equality hold?
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
reducible over the rational numbers, given that it has integer coefficients?
blocks, each weighing at least ; their total weight is 
. Prove that for every real number 
with 
you can choose a subset of the blocks whose total weight is at least but at most 
.
such that 
.
chessboard. Initially, Azer colors all cells of this chessboard with some colors. Then, Babek takes rows and columns and looks at the 
cells in the intersection. Babek wants to have all these cells in a same color, but Azer doesn't. With at least how many colors, Azer can reach his goal?
be non-negative real numbers such that 
Prove that
gangsters in a street.They shoot at each other in the same time.
and 
Find the minimum 
