AoPS Academy
(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
The cases-
isn't an integer
isn't an integer
isn't an integer
which doesn't satisfy
is a positive integer and 
is some integer) being a positive integer, it's possible to get 
. This is an incomplete solution I think.
smallest positive solutions of 
which equals 
,
,
,
,
