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.
is a prime number, 
are natural numbers (
). It is known that 
. Prove that 
.
such that 
and
.
such that for all 
,
, 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
. Find all funcitions 
such that for all any 
be real positive number satisfy that: 
. Prove that: 
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
are positive numbers, then prove that
Prove that
Where 
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
and 
Prove that
has no real roots, and 
takes all real values 
,
.
,
,
--- and so