Difference between revisions of "2006 AIME I Problems/Problem 6"
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| − | Alternatively, for every number, <math>0.\overline{abc}</math>, there will be exactly one other number, such that when they are added together, the sum | + | Alternatively, for every number, <math>0.\overline{abc}</math>, there will be exactly one other number, such that when they are added together, the sum is <math>0.\overline{999}</math>, or, more precisely, 1. As an example, <math>.\overline{123}+.\overline{876}=.\overline{999} \Longrightarrow 1</math>. |
Thus, the solution can be determined by dividing the total number of [[permutation]]s by 2. The answer is <math>\frac{10 \cdot 9 \cdot 8}{2} = \frac{720}{2}= 360</math>. | Thus, the solution can be determined by dividing the total number of [[permutation]]s by 2. The answer is <math>\frac{10 \cdot 9 \cdot 8}{2} = \frac{720}{2}= 360</math>. | ||
Revision as of 17:32, 13 March 2007
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
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 
.
See also
| 2006 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
