2006 AIME I Problems/Problem 6
Let be the set of real numbers that can be represented as repeating decimals of the form where are distinct digits. Find the sum of the elements of
Numbers of the form can be written as . There are such numbers. Each digit will appear in each place value times, and the sum of the digits, 0 through 9, is 45. So the sum of all the numbers is .
Alternatively, for every number, , there will be exactly one other number, such that when they are added together, the sum is , or, more precisely, 1. As an example, .
Thus, the solution can be determined by dividing the total number of permutations by 2. The answer is .
Another method, albeit a little risky, that can be used is to note that the numbers between 1 and 999 with distinct digits average out to . Then the total sum becomes which reduces to
By symmetry, the average over all numbers is . Then, their sum is .
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