AoPS Academy
satisfying 
, prove that![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](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?
+1 w
lie in this order on a line with 
. Let 
be the circles with diameter 
respectively. Points 
and 
and 
are on 
and 
, respectively such that 
. Show that the two lines 
and 
intersect on 
lie in this order on a line with . Let be the circles with diameter respectively. Points and and are on and , respectively such that . Show that the two lines and intersect on 
of the homothety that takes to . Because the centers of these two circles lie on , must lie on too, and the same homothety takes 
to 
and to 
. Since and 
are diameters, 
So 
and since they have two equal angles, triangles 
and 
are similar. Then, the homothety that took segment to segment also took to 
, and the center, , of the homothety is colinear with those two points., that satisfy the condition also satisfy that lies on , as desired.
