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(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
be the incircle of triangle 
. Line 
is parallel to side 
and tangent to . Line 
is parallel to side 
and tangent to . It turned out that the intersection point of and lies on circumcircle of 
satisfy all conditions below:
for all 
for all 
for all
![\[ 4 \geq AC \cdot BD = AB \cdot CD + AD \cdot BC \geq 2 \sqrt{AB \cdot BC \cdot CD \cdot DA} \geq 4 \]](http://latex.artofproblemsolving.com/3/b/b/3bba516895f19c3bf01d208d0554635885b8ceae.png)
(since and 
are not longer than a diameter of the circle). and 
are diameters, 
is a rectangle, and also 
which implies that 
, so is a square, and we're done.
with equality iff and are diameters.
with equality iff 
,
concave on ![$[0, \pi]$](http://latex.artofproblemsolving.com/0/a/1/0a151f5b93ce5735f72cdd79c46051f52b9d820d.png)
?
then AM-GM then jensen.
,
,
,
.
.
and etc, Hence:![\[ AB \cdot BC \cdot CD \cdot DA=16R^2 \cdot \sin{a} \sin{b} \sin{c} \sin{d} \]](http://latex.artofproblemsolving.com/4/a/e/4ae772824bec8a40897191acf9ce19461da38024.png)
.
.
