AoPS Academy
| 
with 
are positive integers and 
is prime (
), we get: 
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 
.
, 
is the midpoint of 
. 
is an arbitrary point on 
. Let 
be the intersection of 
and 
. The line 
intersects with the circumcircle of 
, for the second time at 
. 
is the circumcenter of triangle 
. Prove that 
.
positive such that 
, then there exists 
positive such that 
converges and 
diverges ?
is continuous, positive with 
, does 
continuous and positive exist on 
such that 
converges and 
diverges ?
is in the 
class, increasing : I chose 
. With this idea, I had the idea to define 
but it is not clear that it is ok, even if is increasing. positive such that , then there exists positive such that converges and diverges ? And if is increasing ? positive increasing such that 
, then 
diverges ? positive increasing such that , then 
converges ? is positive increasing and such that 
does not converge to 
, can diverge ? only to be continuous ?
has always integral solutions for 
and 
whenever 
is a positive integer.
and 
.
.
be a function such that 
and for 
Prove that
exists and is less than 
.
satisfied
for all 
and all 
.
. where 
, if 
exists, then 
and by Vieta's formula, we have 
so taking 
, we have 
, so 
for infinitely many , i.e. 
. are integers. Since 
are the roots of , it follows that 
is an integer, as desired.
,
.
and 
![$A = \sum_{r=0}^{24} \frac{d(F(r,x))}{dx} \left[\frac{1}{F(r,x) + \frac{1}{F(r,x)}}\right]$](http://latex.artofproblemsolving.com/4/2/b/42bd0116f69f03884746ba8d06493ed999a567d6.png)
![$B = \sum_{r=0}^{24} \frac{-d(G(r,x))}{dx} \left[\frac{1}{G(r,x) + \frac{1}{G(r,x)}}\right]$](http://latex.artofproblemsolving.com/9/f/5/9f5ff4b5d991249940fa0521b222e5e4201a9fe0.png)
![$R = \frac{A - B}{\tan x\left[PQ - \frac{1}{3^{25}}\right]}$](http://latex.artofproblemsolving.com/b/f/4/bf447429d8f9e71117178abb46c688d7d4a036d6.png)
where 
is divided by 
