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.
and 
Prove that
be an acute triangle and let 
be the midpoint of 
. A circle 
passing through 
and meets the sides 
and 
at points 
and 
respectively. Let 
be the point such that 
is a parallelogram. Suppose that lies on the circumcircle of . Determine all possible values of 
.
satisfying the following property: if 
, 
and 
are integers such that 
, then
such that 
.
be functions 
such that for all reals 
, 
Prove that either 
is the identity function or 
is periodic. from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
be the set of real numbers. Determine all functions such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
from the reals to the reals such that![\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]](http://latex.artofproblemsolving.com/e/b/4/eb4e0d6580d9d3d6a627e39837edb3d1943d4596.png)
.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
such that![\[f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)\]](http://latex.artofproblemsolving.com/9/8/a/98ab64d17bc7d6bf7533b0874177fc963405c1d8.png)
.
