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.
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that for all 
,
is a prime number, 
are natural numbers (
). It is known that 
. Prove that 
.
be the midpoint of 
on triangle 
. Point 
lies on segment 
such that 
and 
on line 
such that 
. Prove that 
.
there exists a unique positive integers 
such that 
is a square of a rational number.
for which![\[2^p = 2^{q-2} + q!.\]](http://latex.artofproblemsolving.com/8/8/2/88270dc4aa4a54cc8b2e525b7d9d940c300e49ac.png)
. Prove that
, segments 
and 
are parallel. Angle bisectors of 
and 
intersect at 
. Circumcircles of 
and 
meet again at 
. Angle bisector of cuts at 
. Prove that
and 
such that 
and 
is a prime number
is natural and 
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid m^n - n.
\]](http://latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
just pick 
, otherwise is coprime to 
. By Chinese Remainder Theorem there exists an such that 
and 
, pick this , we see that both 
and leave remainder 
modulo , so we are done.
are positive numbers, then prove that
Prove that
Where 
