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Difference between revisions of "2014 AMC 12A Problems/Problem 15"

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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>.
  
 +
==Solution 4 (Variation of #2)==
 +
First, allow <math>a</math> to be zero, and then subtract by how much we overcount. We'll also sum each palindrome with its <math>\textit{complement}</math>. If <math>abcba</math> is a palindrome, then its complement is <math>defed</math> where <math>d=9-a</math>, <math>e=9-b</math>, <math>f=9-c</math>. Notice how every palindrome has a unique compliment, and that the sum of a palindrome and its complement is <math>99999</math>. Therefore, the sum of our palindromes is <math>99999\times (10^3/2)</math>. (There are <math>10^3/2</math> pairs.)
 +
 +
However, we have overcounted, as something like <math>05350</math> <math>\textit{isn't}</math> a palindrome by the problem's definition, but we've still included it. So we must subtract the sum of numbers in the form <math>nmn0</math>. By the same argument as before, these sum to <math>9990\times  (10^2/2)</math>. Therefore, the sum that the problem asks for is:
 +
 +
<cmath>500\times99999-50\times 9990</cmath>
 +
<cmath>=500\times99999-500\times 999</cmath>
 +
<cmath>=500(99999-999)</cmath>
 +
<cmath>=500\times 99000</cmath>
 +
 +
Since all we care about is the sum of the digits, we can drop the <math>0</math>'s.
 +
 +
<cmath>5\times99</cmath>
 +
<cmath>=5\times(100-1)</cmath>
 +
<cmath>=495</cmath>
 +
 +
And finally, <math>4+9+5=\boxed{\textbf{(B)}18}</math>
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2014|ab=A|num-b=14|num-a=16}}
 
{{AMC12 box|year=2014|ab=A|num-b=14|num-a=16}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 10:12, 2 February 2021

Problem

A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?

$\textbf{(A) }9\qquad \textbf{(B) }18\qquad \textbf{(C) }27\qquad \textbf{(D) }36\qquad \textbf{(E) }45\qquad$

Solution 1

For each digit $a=1,2,\ldots,9$ there are $10\cdot10$ (ways of choosing $b$ and $c$) palindromes. So the $a$s contribute $(1+2+\cdots+9)(100)(10^4+1)$ to the sum. For each digit $b=0,1,2,\ldots,9$ there are $9\cdot10$ (since $a \neq 0$) palindromes. So the $b$s contribute $(0+1+2+\cdots+9)(90)(10^3+10)$ to the sum. Similarly, for each $c=0,1,2,\ldots,9$ there are $9\cdot10$ palindromes, so the $c$ contributes $(0+1+2+\cdots+9)(90)(10^2)$ to the sum.

It just so happens that \[(1+2+\cdots+9)(100)(10^4+1)+(1+2+\cdots+9)(90)(10^3+10)+(1+2+\cdots+9)(90)(10^2)=49500000\] so the sum of the digits of the sum is $\boxed{\textbf{(B)}\; 18}$.

Solution 2

Notice that $10001+ 99999 = 110000.$ In fact, ordering the palindromes in ascending order, we find that the sum of the nth palindrome and the nth to last palindrome is $110000.$ We have $9*10*10$ palindromes, or $450$ pairs of palindromes summing to $110000.$ Performing the multiplication gives $49500000$, so the sum $\boxed{\textbf{(B)}\; 18}$.

Solution 3

As shown above, there are a total of $900$ 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 $900$ to get our sum. The expected value for the ten-thousands and the units digit is $\frac{1+2+3+\cdots+9}{9}=5$, and the expected value for the thousands, hundreds, and tens digit is $\frac{0+1+2+\cdots+9}{10}=4.5$. Therefore our expected value is $5\times10^4+4.5\times10^3+4.5\times10^2+4.5\times10^1+5\times10^0=55,\!000$. 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 $55,\!000$ or $900$. Thus we only need to calculate $55\times9=495$, and the desired sum is $\boxed{\textbf{(B) }18}$.

Solution 4 (Variation of #2)

First, allow $a$ to be zero, and then subtract by how much we overcount. We'll also sum each palindrome with its $\textit{complement}$. If $abcba$ is a palindrome, then its complement is $defed$ where $d=9-a$, $e=9-b$, $f=9-c$. Notice how every palindrome has a unique compliment, and that the sum of a palindrome and its complement is $99999$. Therefore, the sum of our palindromes is $99999\times (10^3/2)$. (There are $10^3/2$ pairs.)

However, we have overcounted, as something like $05350$ $\textit{isn't}$ a palindrome by the problem's definition, but we've still included it. So we must subtract the sum of numbers in the form $nmn0$. By the same argument as before, these sum to $9990\times (10^2/2)$. Therefore, the sum that the problem asks for is:

\[500\times99999-50\times 9990\] \[=500\times99999-500\times 999\] \[=500(99999-999)\] \[=500\times 99000\]

Since all we care about is the sum of the digits, we can drop the $0$'s.

\[5\times99\] \[=5\times(100-1)\] \[=495\]

And finally, $4+9+5=\boxed{\textbf{(B)}18}$

See Also

2014 AMC 12A (Problems • Answer Key • Resources)
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Problem 14 		Followed by
Problem 16
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All AMC 12 Problems and Solutions

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