There are ten different 10-digit decimal fractions, one of them being equal to the sum of the other nine. If each number has 10 unique digits, not counting the 0 before the decimal point (for example $.9876543210$), what is the least possible value for the one which is the sum of the others?
The answer is
1203456789 (if interpreting as 10-digit integers which can start with zero).
Since each of the numbers is not less than 0123456789, so the sum of the 9 is not less than 123456789*9=1111111101. The smallest number which is not less than 1111111101 and contains all digits is 1203456789. The following set of numbers, for example:
sums up to 1203456789. (This set was found without a computer, roughly with writing down all 123456789's and the remaining part to the desired sum, then swapping the digits which give equal sums to make all numbers different.)