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
Let 
be an acute triangle and let 
be its circumcenter. Let 
be the feet of the altitudes from 
, and 
the midpoints of the sides 
, 
, 
, respectively. Consider the circle 
that passes through and is tangent to at 
. Similarly, 
and 
are defined. Let 
be the intersection of and , other than . Similarly, 
and 
are defined. Prove that 
, 
and 
concur in the centroid of triangle .
be an acute triangle and let 
be its circumcenter. Let 
be the feet of the altitudes from 
, and 
the midpoints of the sides 
, 
, 
, respectively. Consider the circle 
that passes through and is tangent to at 
. Similarly, 
and 
are defined. Let 
be the intersection of and , other than . Similarly, 
and 
are defined. Prove that 
, 
and 
concur in the centroid of triangle .
