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 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.
such that 
.
is defined by 
and for all positive integers 
, 
. Show that for every positive integer 
, there exists a suitable positive integer 
such that 
is a multiple of .
. Let 
be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
be an odd prime. Define 
over the positive integers as follows:
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
be the nonnegative integers in 
where 
. Define 
. Prove that 
. such that 
where 
, and the statement is still true when 
? such that 
where 
, and the statement is still true when ?.
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and are integers and is a prime number.
such that 
divides 
.
has as its vertices the feet of the altitudes of a non-right triangle 
. Find all possibilities for the sizes of the angles of for which the triangles and are similar.
.
.