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.
Compute 
be an infinite sequence and 
be a positive integer such that 
, and 
for all 
. Prove that 
is a perfect square for all 
.
such that for all real 
![\[f(x+f(y))=f(x+y)+y\]](http://latex.artofproblemsolving.com/1/e/2/1e2fef35ee53cd3224527f672b9edbb7a5293fad.png)
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
write all its divisors in increasing order: 
. such that 
.
such that both 
and 
are divisible by 
.
+1 w
, there exist integers 
such that, if we let 
, then the following conditions hold:
we have 
; the number 
is divisible by ; and the numbers 
each have a prime factor not dividing .
, and 
. The spheres with centers , and touch the plane at 
, and 
, respectively. Prove that 
and 
Multiplying all such equations yields
which reduces to your first identity.
and apply the AM-GM:![$$(C_1P_1)^2+(C_2P_2)^2+(C_3P_3)^2+\frac1{(C_1P_1)^2}+\frac1{(C_2P_2)^2}+\frac1{(C_3P_3)^2}\ge6\sqrt[6]1=12.$$](http://latex.artofproblemsolving.com/7/8/a/78a681d09dd29693e349f8661fdf417070d87b9a.png)