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.
be positive integers such that one of the values 
is equal to the product of the remaining two. Prove that one of the numbers is a multiple of another of them.
and 
is entered, the new password is 
, which is determined by and (naturally, the person does not know or ). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.
interesting if 
is composite and for every divisor 
of at least one of 
and 
is also a divisor of with 
such that 
and 
is a perfect square.
has a right angle at 
. Let 
be the point on line 
such that 
and 
lies between and 
. Point 
is chosen so that 
and 
is the bisector of 
. Point 
is chosen so that 
and 
is the bisector of 
. Let 
be the midpoint of . Let 
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
of the convex quadrilateral 
be such that 
Let 
be the center of the circumcircle of the triangle 
. Prove that 
.
such that 
. Prove the following holds
and 
.
Prove 
Find maximum 
