Difference between revisions of "2014 AMC 12A Problems/Problem 15"
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Notice that <math>10001+ 99999 = 110000.</math> In fact, ordering the palindromes in ascending order, we find that the sum of the nth palindrome and the nth to last palindrome is <math>110000.</math> We have <math>9*10*10</math> palindromes, or <math>450</math> pairs of palindromes summing to <math>110000.</math> Performing the multiplication gives <math>49500000</math>, so the sum is <math>18</math> <math> \boxed{\textbf{(B)}} </math>. | Notice that <math>10001+ 99999 = 110000.</math> In fact, ordering the palindromes in ascending order, we find that the sum of the nth palindrome and the nth to last palindrome is <math>110000.</math> We have <math>9*10*10</math> palindromes, or <math>450</math> pairs of palindromes summing to <math>110000.</math> Performing the multiplication gives <math>49500000</math>, so the sum is <math>18</math> <math> \boxed{\textbf{(B)}} </math>. | ||
− | ==Solution | + | ==Solution 3== |
As shown above, there are a total of <math>900</math> five-digit palindromes. We can calculate their sum by finding the expected value of a randomly selected palindrome satisfying the conditions given, then multiplying it by <math>900</math> to get our sum. The expected value for the ten-thousands and the units digit is <math>\frac{1+2+3+\cdots+9}{9}=5</math>, and the expected value for the thousands, hundreds, and tens digit is <math>\frac{0+1+2+\cdots+9}{10}=4.5</math>. Therefore our expected value is <math>5\times10^4+4.5\times10^3+4.5\times10^2+4.5\times10^1+5\times10^0=55,\!000</math>. Since the question asks for the sum of the digits of the resulting sum, we do not need to keep the trailing zeros of either <math>55,\!000</math> or <math>900</math>. Thus we only need to calculate <math>55\times9=495</math>, and the desired sum is <math>\boxed{\textbf{(B) }18}</math>. | As shown above, there are a total of <math>900</math> five-digit palindromes. We can calculate their sum by finding the expected value of a randomly selected palindrome satisfying the conditions given, then multiplying it by <math>900</math> to get our sum. The expected value for the ten-thousands and the units digit is <math>\frac{1+2+3+\cdots+9}{9}=5</math>, and the expected value for the thousands, hundreds, and tens digit is <math>\frac{0+1+2+\cdots+9}{10}=4.5</math>. Therefore our expected value is <math>5\times10^4+4.5\times10^3+4.5\times10^2+4.5\times10^1+5\times10^0=55,\!000</math>. Since the question asks for the sum of the digits of the resulting sum, we do not need to keep the trailing zeros of either <math>55,\!000</math> or <math>900</math>. Thus we only need to calculate <math>55\times9=495</math>, and the desired sum is <math>\boxed{\textbf{(B) }18}</math>. | ||
Revision as of 14:13, 17 March 2015
Contents
[hide]Problem
A five-digit palindrome is a positive integer with respective digits , where is non-zero. Let be the sum of all five-digit palindromes. What is the sum of the digits of ?
Solution 1
For each digit there are (ways of choosing and ) palindromes. So the s contribute to the sum. For each digit there are (since ) palindromes. So the s contribute to the sum. Similarly, for each there are palindromes, so the contributes to the sum.
It just so happens that so the sum of the digits of the sum is , or .
Solution 2
Notice that In fact, ordering the palindromes in ascending order, we find that the sum of the nth palindrome and the nth to last palindrome is We have palindromes, or pairs of palindromes summing to Performing the multiplication gives , so the sum is .
Solution 3
As shown above, there are a total of five-digit palindromes. We can calculate their sum by finding the expected value of a randomly selected palindrome satisfying the conditions given, then multiplying it by to get our sum. The expected value for the ten-thousands and the units digit is , and the expected value for the thousands, hundreds, and tens digit is . Therefore our expected value is . Since the question asks for the sum of the digits of the resulting sum, we do not need to keep the trailing zeros of either or . Thus we only need to calculate , and the desired sum is .
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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