2006 AIME I Problems/Problem 6
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[hide]Problem
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
Solution 1
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 .
Solution 2
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
See also
2006 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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