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 a primes ,
be positive integers such that 
and 
.
+1 w
satisfying 
, find the biggest value the following equation could acquire:
be the positive number satisfied 
Find the minimum of 
be positive integers such that no number divides some other number. If 
, prove that 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
. There are 
mutually non-attacking rooks placed on a grid 
. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?
such that the largest prime divisor of 
is equal to the largest prime divisor of 
.
points inside a circle of radius 
Show that there are at least 
of these points that are inside a circle of radius 
the area of a circle radius 
pigeons
must have at least 
points inside
be real numbers such that 
.
