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

Latest revision as of 21:42, 2 August 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 $9\cdot 10\cdot 10$ 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 $\overline{abcba}$ (the line means a, b, and c are digits and $abcba\ne a\cdot b\cdot c\cdot b\cdot a$ ) is a palindrome, then its complement is $\overline{defed}$ 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 $\overline{0nmn0}$ . 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, $4+9+5=\boxed{\textbf{(B)}18}$

@@ Line 9: / Line 9: @@
 \textbf{(E) }45\qquad</math>
-==Solution One==
+==Solution 1==
 For each digit <math>a=1,2,\ldots,9</math> there are <math>10\cdot10</math> (ways of choosing <math>b</math> and <math>c</math>) palindromes. So the <math>a</math>s contribute <math>(1+2+\cdots+9)(100)(10^4+1)</math> to the sum.
@@ Line 15: / Line 15: @@
 Similarly, for each <math>c=0,1,2,\ldots,9</math> there are <math>9\cdot10</math> palindromes, so the <math>c</math> contributes <math>(0+1+2+\cdots+9)(90)(10^2)</math> to the sum.
-It just so happens that <cmath> (1+2+\cdots+9)(100)(10^4+1)+(1+2+\cdots+9)(90)(10^3+10)+(1+2+\cdots+9)(90)(10^2)=49500000 </cmath> so the sum of the digits of the sum is <math>18</math>, or <math>\boxed{\textbf{(B)}}</math>.
+It just so happens that <cmath> (1+2+\cdots+9)(100)(10^4+1)+(1+2+\cdots+9)(90)(10^3+10)+(1+2+\cdots+9)(90)(10^2)=49500000 </cmath> so the sum of the digits of the sum is <math>\boxed{\textbf{(B)}\; 18}</math>.
-(Solution by AwesomeToad)
+==Solution 2==
+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\cdot 10\cdot 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 <math> \boxed{\textbf{(B)}\; 18}</math>.
-==Solution Two==
+==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 there are only <math>9\cdot10\cdot10 = 900</math> five digit palindromes, it is sufficient to add up all of them.
+==Solution 4 (Variation of #2)==
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+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>\overline{abcba}</math> (the line means a, b, and c are digits and <math>abcba\ne a\cdot b\cdot c\cdot b\cdot a</math>) is a palindrome, then its complement is <math>\overline{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.)
-==Solution Three==
+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>\overline{0nmn0}</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:
-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>
+<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==
 {{AMC12 box|year=2014|ab=A|num-b=14|num-a=16}}
 {{MAA Notice}}

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