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 Adventure10, Mango247
The circles 
and 
are externally tangent in pairs (each tangent to other two externally). Let 
the common point of 
and 
the common point of 
and and 
the common point of and . Let 
be an arbitrary point of . Line 
cuts in 
, line 
cuts in 
and line 
cuts in 
. Prove that 
is diameter of .
and 
are externally tangent in pairs (each tangent to other two externally). Let 
the common point of 
and 
the common point of 
and and 
the common point of and . Let 
be an arbitrary point of . Line 
cuts in 
, line 
cuts in 
and line 
cuts in 
. Prove that 
is diameter of .
and 
are centers of circles. With homotheties centered at 
we have 
.
