rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to 
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 guangzhou-2015, Adventure10
An convex qudrilateral 
is given. 
is the intersection point of the diagonals 
and 
. The points 
lie respectively on 
such that 
.
The circumcircles of
and 
meet at second time at 
and the the circumcircles of 
and 
- at 
.
The circumcircles of
and 
meet at second time at 
and the the circumcircles of 
and 
- at 
.
Prove that the quadrilateral
is cyclic.
is given. 
is the intersection point of the diagonals 
and 
. The points 
lie respectively on 
such that 
.The circumcircles of
and 
meet at second time at 
and the the circumcircles of 
and 
- at 
.The circumcircles of
and 
meet at second time at 
and the the circumcircles of 
and 
- at 
.Prove that the quadrilateral
is cyclic.
cuts 
at 
and 
is cyclic.
and 
are similar, then two triangles 
and 
are similar, then 
and 
be the midpoints of 
passes through four points 
.
