be an odd integer. Prove that there exists a prime number 
such that![\[
p \mid 2^{\varphi(n)} - 1 \quad \text{but} \quad p \nmid n.
\]](http://latex.artofproblemsolving.com/3/5/1/3514b64ecd5f7b2586d5bfeae59b7c9504d5b455.png)
such that 
.
and 
meet the sides 
and 
at 
and 
respectively. Prove that ![\[ AB+AC\geq4DE. \]](http://latex.artofproblemsolving.com/3/a/4/3a49c5f8bd2930adca7bde2601dd045d4bf05ff6.png)
such that ![\[f(xf(y)+f(x+y)+1)=(y+1)f(x+1)\]](http://latex.artofproblemsolving.com/4/9/e/49ec422b640766de653d9d640bbc45065a6ac375.png)
holds for all 
.
on one variable with real coefficients such that the equation 
has exactly 
ordered pairs 
as solutions?
Y by Adventure10, Mango247
In triangle 
, 
, 
and 
are the circumcenter and incenter respectively, Points 
and 
, such that 
. Prove that 
and 
.
, 
, 
and 
are the circumcenter and incenter respectively, Points 
and 
, such that 
. Prove that 
and 
.
