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.
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
. How many cats of each breed does he have?
satisfying 
Prove that 
be positive integers. If for each positive integer 
we have that 
Prove that 
.
, if any element of the set 
is fantabulous, then all the elements are fantabulous. Does it follow that the number 
is fantabulous?
, in which 
: 
, but the proof is important.
is a cyclic quadrilateral with 
. Let 
and 
be the feet of perpendiculars from 
and 
to 
respectively. Prove that 
.
consisting of a prime number 
and two positive integers 
and 
such that 
and 
are both powers of .
be integers. Suppose that for each 
there exists an integer 
such that 
is divisible by 
. Prove that 
for some integer .
, find the maximal area of the triangle.
on the whiteboard. In each move, she selects two distinct numbers, 
and 
, erases them, and replaces them with 
. She repeats this process until only one number, , remains. What are all the possible values of ?
, 
is a perfect square.
, so that the number 
is a prime.
be a convex polygon. For 
, denote by 
the internal angle at vertex 
. For any point 
inside 
, prove that![\[
\cos\frac{\alpha_1}{2}PA_1 + \cos\frac{\alpha_2}{2}PA_2 + \ldots + \cos\frac{\alpha_n}{2}PA_n \ge s,
\]](http://latex.artofproblemsolving.com/0/4/0/040ab0ab9622c1de8be2d45af3755df854565f14.png)
where 
denotes the semi-perimeter of .
and 
Prove that
