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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?

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  • $\begingroup$ I'm having a bit of difficulty understanding this "one that is equals the sum of all the other decimal numbers" - do you mean that, among the ten numbers, one of them equals the sum of the other nine? By "decimal number", do you mean a large number (several billion) expressed in base 10, or a small number (between 0 and 1) expressed as a decimal fraction? $\endgroup$ – Rand al'Thor Jan 4 at 11:34
  • $\begingroup$ @Rand al'thor -right one of them is the sum of the other nine and these decimal numbers are less than 1. $\endgroup$ – TSLF Jan 4 at 11:47
  • $\begingroup$ lets ignore the zero before the decimal point $\endgroup$ – TSLF Jan 4 at 11:49
  • $\begingroup$ Are you taking the sum modulo 1 (i.e. ignoring everything before the decimal point), or is the sum of those nine 10-digit numbers directly equal to the tenth one? $\endgroup$ – Rand al'Thor Jan 4 at 12:00
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    $\begingroup$ It might be simpler to think of them as 10-digit integers that are allowed to have leading zeros. $\endgroup$ – Gareth McCaughan Jan 4 at 13:55
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The answer is

1203456789 (if interpreting as 10-digit integers which can start with zero).

Reasoning

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:
a=0124635789
b=0123456789
c=0123756489
d=0123457689
e=0123546789
f=0182345679
g=0152364789
h=0126435789
i=0123456987
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.)

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