satisfying 
,![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
(use in a problem for junior) to practice. Can anyone share to me?
Y by oolite
Let 
be a triangle. Let 
and 
be points on the side 
such that lies on the segment 
and 
. Similarly, let 
and 
be points on the side 
such that lies on the segment 
and 
. The segments 
and 
meet at 
, and the circumcircles of 
and 
meet again at S, inside triangle 
. Finally, let 
be the midpoint of the side 
. Prove that the angles 
and 
are equal.
be a triangle. Let 
and 
be points on the side 
such that lies on the segment 
and 
. Similarly, let 
and 
be points on the side 
such that lies on the segment 
and 
. The segments 
and 
meet at 
, and the circumcircles of 
and 
meet again at S, inside triangle 
. Finally, let 
be the midpoint of the side 
. Prove that the angles 
and 
are equal.
