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'ThorJan 4, 2020 at 11:34
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$\begingroup$ @Rand al'thor -right one of them is the sum of the other nine and these decimal numbers are less than 1. $\endgroup$– TSLFJan 4, 2020 at 11:47
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$\begingroup$ lets ignore the zero before the decimal point $\endgroup$– TSLFJan 4, 2020 at 11:49
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$\begingroup$ 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'ThorJan 4, 2020 at 12:00
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1$\begingroup$ It might be simpler to think of them as 10-digit integers that are allowed to have leading zeros. $\endgroup$– Gareth McCaughan ♦Jan 4, 2020 at 13:55
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1 Answer
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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.)
answered Jan 4, 2020 at 22:44