2014 AMC 12A Problems/Problem 15

Revision as of 14:52, 25 January 2015 by Pi31415926535897 (talk | contribs) (Solution Two)

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 One

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 $18$, or $\boxed{\textbf{(B)}}$.

(Solution by AwesomeToad)

Solution Two

As there are only $9\cdot10\cdot10 = 900$ five digit palindromes, it is sufficient to add up all of them. \[10001 + 10101 + 10201 + 10301 + 10401 + 10501 + 10601 + 10701 + 10801 + 10901 + 11011 + 11111 + 11211 + 11311 + 11411 + 11511 + 11611 + 11711 + 11811 + 11911 + 12021 + 12121 + 12221 + 12321 + 12421 + 12521 + 12621 + 12721 + 12821\]\[+ 12921 + 13031 + 13131 + 13231 + 13331 + 13431 + 13531 + 13631 + 13731 + 13831 + 13931 + 14041 + 14141 + 14241 + 14341 + 14441 + 14541 + 14641 + 14741 + 14841 + 14941 + 15051 + 15151 + 15251 + 15351 + 15451 + 15551 + 15651\]\[+ 15751 + 15851 + 15951 + 16061 + 16161 + 16261 + 16361 + 16461 + 16561 + 16661 + 16761 + 16861 + 16961 + 17071 + 17171 + 17271 + 17371 + 17471 + 17571 + 17671 + 17771 + 17871 + 17971 + 18081 + 18181 + 18281 + 18381 + 18481\]\[+ 18581 + 18681 + 18781 + 18881 + 18981 + 19091 + 19191 + 19291 + 19391 + 19491 + 19591 + 19691 + 19791 + 19891 + 19991 + 20002 + 20102 + 20202 + 20302 + 20402 + 20502 + 20602 + 20702 + 20802 + 20902 + 21012 + 21112 + 21212\]\[+ 21312 + 21412 + 21512 + 21612 + 21712 + 21812 + 21912 + 22022 + 22122 + 22222 + 22322 + 22422 + 22522 + 22622 + 22722 + 22822 + 22922 + 23032 + 23132 + 23232 + 23332 + 23432 + 23532 + 23632 + 23732 + 23832 + 23932 + 24042\]\[+ 24142 + 24242 + 24342 + 24442 + 24542 + 24642 + 24742 + 24842 + 24942 + 25052 + 25152 + 25252 + 25352 + 25452 + 25552 + 25652 + 25752 + 25852 + 25952 + 26062 + 26162 + 26262 + 26362 + 26462 + 26562 + 26662 + 26762 + 26862\]\[+ 26962 + 27072 + 27172 + 27272 + 27372 + 27472 + 27572 + 27672 + 27772 + 27872 + 27972 + 28082 + 28182 + 28282 + 28382 + 28482 + 28582 + 28682 + 28782 + 28882 + 28982 + 29092 + 29192 + 29292 + 29392 + 2:)9492 + 29592 + 29692\]\[+ 29792 + 29892 + 29992 + 30003 + 30103 + 30203 + 30303 + 30403 + 30503 + 30603 + 30703 + 30803 + 30903 + 31013 + 31113 + 31213 + 31313 + 31413 + 31513 + 31613 + 31713 + 31813 + 31913 + 32023 + 32123 + 32223 + 32323 + 32423\]\[+ 32523 + 32623 + 32723 + 32823 + 32923 + 33033 + 33133 + 33233 + 33333 + 33433 + 33533 + 33633 + 33733 + 33833 + 33933 + 34043 + 34143 + 34243 + 34343 + 34443 + 34543 + 34643 + 34743 + 34843 + 34943 + 35053 + 35153 + 35253\]\[+ 35353 + 35453 + 35553 + 35653 + 35753 + 35853 + 35953 + 36063 + 36163 + 36263 + 36363 + 36463 + 36563 + 36663 + 36763 + 36863 + 36963 + 37073 + 37173 + 37273 + 37373 + 37473 + 37573 + 37673 + 37773 + 37873 + 37973 + 38083\]\[+ 38183 + 38283 + 38383 + 38483 + 38583 + 38683 + 38783 + 38883 + 38983 + 39093 + 39193 + 39293 + 39393 + 39493 + 39593 + 39693 + 39793 + 39893 + 39993 + 40004 + 40104 + 40204 + 40304 + 40404 + 40504 + 40604 + 40704 + 40804\]\[+ 40904 + 41014 + 41114 + 41214 + 41314 + 41414 + 41514 + 41614 + 41714 + 41814 + 41914 + 42024 + 42124 + 42224 + 42324 + 42424 + 42524 + 42624 + 42724 + 42824 + 42924 + 43034 + 43134 + 43234 + 43334 + 43434 + 43534 + 43634\]\[+ 43734 + 43834 + 43934 + 44044 + 44144 + 44244 + 44344 + 44444 + 44544 + 44644 + 44744 + 44844 + 44944 + 45054 + 45154 + 45254 + 45354 + 45454 + 45554 + 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94949 + 95059 + 95159 + 95259 + 95359 + 95459 + 95559 + 95659 + 95759 + 95859 + 95959 + 96069 + 96169 + 96269 + 96369 + 96469 + 96569 + 96669 + 96769 + 96869\]\[+ 96969 + 97079 + 97179 + 97279 + 97379 + 97479 + 97579 + 97679 + 97779 + 97879 + 97979 + 98089 + 98189 + 98289 + 98389 + 98489 + 98589 + 98689 + 98789 + 98889 + 98989 + 99099 + 99199 + 99299 + 99399 + 99499 + 99599 + 99699\]\[+ 99799 + 99899 + 99999 = 49500000 = 4 + 9 + 5 = 18 \to \boxed{(B)}\].

Solution Three

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 is $18$ $\boxed{\textbf{(B)}}$.

See Also

2014 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 12 Problems and Solutions

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