Question 8845
 First of allyou posted the question in wrong category.

 The number 5abc6 > 50006 ,since sqrt(50006)~ 223.6202138
 And we know that if the last digit of a perfect square x^2 is 6 then 
 the last digit of x must be 4 or 6.

 Starting from 224, we try as the table below:

 x       x^2
--------------
 224	50176	a=0, b =1, c = 7,a+b+c = 8	
 226	51076	No, violates to a < b < c	
 234	54756	No, violates to a < b < c	
 236	55696	OK,a+b+c = 20	
 244	59536	No, violates to a < b < c	
 246	60516	Stop,too big (the first digit > 5)

 Hence, the possible values of a+b+c = 8 or 20.

 In my opinion, it should belong to MS Excel,and not a real math
 question.

 Kenny