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 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)
.
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
(so 
is a function from the positive real numbers) such that![\[ \frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}
\]](http://latex.artofproblemsolving.com/0/4/b/04b955fb570ad384fb7f0e85bf0dacccf4c982a1.png)
satisfying 
such that for all integers and , ![\[ f(2x+f(y))+f(f(2x))=y. \]](http://latex.artofproblemsolving.com/b/0/d/b0d16e87a8b30ebec5997ed12254094cdc3125d5.png)
such that 
for all 
be the set of integers. Determine all functions such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
(from the set of real numbers to itself) where
for all reals 
:
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
be the altitude and 
be the intersection point of the altitudes of triangle 
. Point 
is symmetric to with respect to 
. The circumcircle of triangle 
intersects at points 
and 
. Prove that 
.
.
?
![\[
|z^2+1|^2-4|z|^2=\left(|z-i|^2-2\right)\left(|z+i|^2-2\right),
\]](http://latex.artofproblemsolving.com/a/c/a/acaddd001cbb5caa421ee3ce671395b3eb1b598c.png)
thus the locus consists of two circles.
,
,![\[
|z^2 + 1|^2 = 4|z|^2.
\]](http://latex.artofproblemsolving.com/8/e/d/8ed9109f5617a70db4bf88c4ccc7502d81d85449.png)
,![\[
4|z|^2 = 2(|z - i|^2 + |z + i|^2 - 2),
\]](http://latex.artofproblemsolving.com/1/1/9/11964c869076a4e7829903c62834d1c0ebe829a8.png)
![\[
|z^2 + 1|^2 - 2|z - i|^2 - 2|z + i|^2 + 4 = 0.
\]](http://latex.artofproblemsolving.com/7/6/8/7688fd7a1f4fcce5467726981027b37948a23798.png)
,![\[
(|z - i|^2 - 2)(|z + i|^2 - 2) = 0.
\]](http://latex.artofproblemsolving.com/0/3/d/03d088bccc33a9abdab44fff0ac9151f2a50b6a7.png)
and centers 
.