,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that the equation
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
Y by Adventure10, Mango247
In a circle 
with centre 
and radius 
, let 
, 
be two circles with centres 
, 
and radii 
, 
respectively, so that each circle 
is internally tangent to at 
and so that , are externally tangent to each other at 
.
Prove that the three lines
, 
, and 
are concurrent.
with centre 
and radius 
, let 
, 
be two circles with centres 
, 
and radii 
, 
respectively, so that each circle 
is internally tangent to at 
and so that , are externally tangent to each other at 
.Prove that the three lines
, 
, and 
are concurrent.
to 
,
.
to 
balance at 
to 
,
balance at 
at 
.
now balance at 
,
in the triangle 
are concurrent.
,
,
in 
,
so 
are on same line 
a
are on same line.
is perpendicular to line that tangent to circl
is perpendicular to line that tangent to circl
are on same line 
!
and 
and 
AND 
we must proof:
a
so the problem is proof!
.![\[\frac{OA_1}{A_1O_1} \cdot \frac{O_1A}{AO_2} \cdot \frac{O_2A_2}{A_2O} = 1.\]](http://latex.artofproblemsolving.com/5/8/f/58fe3d80b59713a430f2bccd0d008e182f16eb4c.png)
![\[\frac{r}{r_1} \cdot \frac{r_1}{r_2} \cdot \frac{r_2}{r} = 1,\]](http://latex.artofproblemsolving.com/6/d/9/6d92534ae3bab68d714a138855a202469eb1b2e7.png)
