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

Revision as of 10:14, 16 February 2014

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.

$10001 + 10101 + 10201 + 10301 + 10401 + 10501 + 10601 + 10701 + 10801 + 10901 + 11011 + 11111 + 11211 + 11311 + 11411 + 11511 + 11611 + 11711 + 11811 + 11911 + 12021 + 12121 + 12221 + 12321 + 12421 + 12521 + 12621 + 12721 + 12821$ $+ 12921 + 13031 + 13131 + 13231 + 13331 + 13431 + 13531 + 13631 + 13731 + 13831 + 13931 + 14041 + 14141 + 14241 + 14341 + 14441 + 14541 + 14641 + 14741 + 14841 + 14941 + 15051 + 15151 + 15251 + 15351 + 15451 + 15551 + 15651$ $+ 15751 + 15851 + 15951 + 16061 + 16161 + 16261 + 16361 + 16461 + 16561 + 16661 + 16761 + 16861 + 16961 + 17071 + 17171 + 17271 + 17371 + 17471 + 17571 + 17671 + 17771 + 17871 + 17971 + 18081 + 18181 + 18281 + 18381 + 18481$ $+ 18581 + 18681 + 18781 + 18881 + 18981 + 19091 + 19191 + 19291 + 19391 + 19491 + 19591 + 19691 + 19791 + 19891 + 19991 + 20002 + 20102 + 20202 + 20302 + 20402 + 20502 + 20602 + 20702 + 20802 + 20902 + 21012 + 21112 + 21212$ $+ 21312 + 21412 + 21512 + 21612 + 21712 + 21812 + 21912 + 22022 + 22122 + 22222 + 22322 + 22422 + 22522 + 22622 + 22722 + 22822 + 22922 + 23032 + 23132 + 23232 + 23332 + 23432 + 23532 + 23632 + 23732 + 23832 + 23932 + 24042$ $+ 24142 + 24242 + 24342 + 24442 + 24542 + 24642 + 24742 + 24842 + 24942 + 25052 + 25152 + 25252 + 25352 + 25452 + 25552 + 25652 + 25752 + 25852 + 25952 + 26062 + 26162 + 26262 + 26362 + 26462 + 26562 + 26662 + 26762 + 26862$ $+ 26962 + 27072 + 27172 + 27272 + 27372 + 27472 + 27572 + 27672 + 27772 + 27872 + 27972 + 28082 + 28182 + 28282 + 28382 + 28482 + 28582 + 28682 + 28782 + 28882 + 28982 + 29092 + 29192 + 29292 + 29392 + 29492 + 29592 + 29692$ $+ 29792 + 29892 + 29992 + 30003 + 30103 + 30203 + 30303 + 30403 + 30503 + 30603 + 30703 + 30803 + 30903 + 31013 + 31113 + 31213 + 31313 + 31413 + 31513 + 31613 + 31713 + 31813 + 31913 + 32023 + 32123 + 32223 + 32323 + 32423$ $+ 32523 + 32623 + 32723 + 32823 + 32923 + 33033 + 33133 + 33233 + 33333 + 33433 + 33533 + 33633 + 33733 + 33833 + 33933 + 34043 + 34143 + 34243 + 34343 + 34443 + 34543 + 34643 + 34743 + 34843 + 34943 + 35053 + 35153 + 35253$ $+ 35353 + 35453 + 35553 + 35653 + 35753 + 35853 + 35953 + 36063 + 36163 + 36263 + 36363 + 36463 + 36563 + 36663 + 36763 + 36863 + 36963 + 37073 + 37173 + 37273 + 37373 + 37473 + 37573 + 37673 + 37773 + 37873 + 37973 + 38083$ $+ 38183 + 38283 + 38383 + 38483 + 38583 + 38683 + 38783 + 38883 + 38983 + 39093 + 39193 + 39293 + 39393 + 39493 + 39593 + 39693 + 39793 + 39893 + 39993 + 40004 + 40104 + 40204 + 40304 + 40404 + 40504 + 40604 + 40704 + 40804$ $+ 40904 + 41014 + 41114 + 41214 + 41314 + 41414 + 41514 + 41614 + 41714 + 41814 + 41914 + 42024 + 42124 + 42224 + 42324 + 42424 + 42524 + 42624 + 42724 + 42824 + 42924 + 43034 + 43134 + 43234 + 43334 + 43434 + 43534 + 43634$ $+ 43734 + 43834 + 43934 + 44044 + 44144 + 44244 + 44344 + 44444 + 44544 + 44644 + 44744 + 44844 + 44944 + 45054 + 45154 + 45254 + 45354 + 45454 + 45554 + 45654 + 45754 + 45854 + 45954 + 46064 + 46164 + 46264 + 46364 + 46464$ $+ 46564 + 46664 + 46764 + 46864 + 46964 + 47074 + 47174 + 47274 + 47374 + 47474 + 47574 + 47674 + 47774 + 47874 + 47974 + 48084 + 48184 + 48284 + 48384 + 48484 + 48584 + 48684 + 48784 + 48884 + 48984 + 49094 + 49194 + 49294$ $+ 49394 + 49494 + 49594 + 49694 + 49794 + 49894 + 49994 + 50005 + 50105 + 50205 + 50305 + 50405 + 50505 + 50605 + 50705 + 50805 + 50905 + 51015 + 51115 + 51215 + 51315 + 51415 + 51515 + 51615 + 51715 + 51815 + 51915 + 52025$ $+ 52125 + 52225 + 52325 + 52425 + 52525 + 52625 + 52725 + 52825 + 52925 + 53035 + 53135 + 53235 + 53335 + 53435 + 53535 + 53635 + 53735 + 53835 + 53935 + 54045 + 54145 + 54245 + 54345 + 54445 + 54545 + 54645 + 54745 + 54845$ $+ 54945 + 55055 + 55155 + 55255 + 55355 + 55455 + 55555 + 55655 + 55755 + 55855 + 55955 + 56065 + 56165 + 56265 + 56365 + 56465 + 56565 + 56665 + 56765 + 56865 + 56965 + 57075 + 57175 + 57275 + 57375 + 57475 + 57575 + 57675$ $+ 57775 + 57875 + 57975 + 58085 + 58185 + 58285 + 58385 + 58485 + 58585 + 58685 + 58785 + 58885 + 58985 + 59095 + 59195 + 59295 + 59395 + 59495 + 59595 + 59695 + 59795 + 59895 + 59995 + 60006 + 60106 + 60206 + 60306 + 60406$ $+ 60506 + 60606 + 60706 + 60806 + 60906 + 61016 + 61116 + 61216 + 61316 + 61416 + 61516 + 61616 + 61716 + 61816 + 61916 + 62026 + 62126 + 62226 + 62326 + 62426 + 62526 + 62626 + 62726 + 62826 + 62926 + 63036 + 63136 + 63236$ $+ 63336 + 63436 + 63536 + 63636 + 63736 + 63836 + 63936 + 64046 + 64146 + 64246 + 64346 + 64446 + 64546 + 64646 + 64746 + 64846 + 64946 + 65056 + 65156 + 65256 + 65356 + 65456 + 65556 + 65656 + 65756 + 65856 + 65956 + 66066$ $+ 66166 + 66266 + 66366 + 66466 + 66566 + 66666 + 66766 + 66866 + 66966 + 67076 + 67176 + 67276 + 67376 + 67476 + 67576 + 67676 + 67776 + 67876 + 67976 + 68086 + 68186 + 68286 + 68386 + 68486 + 68586 + 68686 + 68786 + 68886$ $+ 68986 + 69096 + 69196 + 69296 + 69396 + 69496 + 69596 + 69696 + 69796 + 69896 + 69996 + 70007 + 70107 + 70207 + 70307 + 70407 + 70507 + 70607 + 70707 + 70807 + 70907 + 71017 + 71117 + 71217 + 71317 + 71417 + 71517 + 71617$ $+ 71717 + 71817 + 71917 + 72027 + 72127 + 72227 + 72327 + 72427 + 72527 + 72627 + 72727 + 72827 + 72927 + 73037 + 73137 + 73237 + 73337 + 73437 + 73537 + 73637 + 73737 + 73837 + 73937 + 74047 + 74147 + 74247 + 74347 + 74447$ $+ 74547 + 74647 + 74747 + 74847 + 74947 + 75057 + 75157 + 75257 + 75357 + 75457 + 75557 + 75657 + 75757 + 75857 + 75957 + 76067 + 76167 + 76267 + 76367 + 76467 + 76567 + 76667 + 76767 + 76867 + 76967 + 77077 + 77177 + 77277$ $+ 77377 + 77477 + 77577 + 77677 + 77777 + 77877 + 77977 + 78087 + 78187 + 78287 +78387 + 78487 + 78587 + 78687 + 78787 + 78887 + 78987 + 79097 + 79197 + 79297 + 79397 + 79497 + 79597 + 79697 + 79797 + 79897 + 79997 + 80008$ $+ 80108 + 80208 + 80308 + 80408 + 80508 + 80608 + 80708 + 80808 + 80908 + 81018 + 81118 + 81218 + 81318 + 81418 + 81518 + 81618 + 81718 + 81818 + 81918 + 82028 + 82128 + 82228 + 82328 + 82428 + 82528 + 82628 + 82728 + 82828$ $+ 82928 + 83038 + 83138 + 83238 + 83338 + 83438 + 83538 + 83638 + 83738 + 83838 + 83938 + 84048 + 84148 + 84248 + 84348 + 84448 + 84548 + 84648 + 84748 + 84848 + 84948 + 85058 + 85158 + 85258 + 85358 + 85458 + 85558 + 85658$ $+ 85758 + 85858 + 85958 + 86068 + 86168 + 86268 + 86368 + 86468 + 86568 + 86668 + 86768 + 86868 + 86968 + 87078 + 87178 + 87278 + 87378 + 87478 + 87578 + 87678 + 87778 + 87878 + 87978 + 88088 + 88188 + 88288 + 88388 + 88488$ $+ 88588 + 88688 + 88788 + 88888 + 88988 + 89098 + 89198 + 89298 + 89398 + 89498 + 89598 + 89698 + 89798 + 89898 + 89998 + 90009 + 90109 + 90209 + 90309 + 90409 + 90509 + 90609 + 90709 + 90809 + 90909 + 91019 + 91119 + 91219$ $+ 91319 + 91419 + 91519 + 91619 + 91719 + 91819 + 91919 + 92029 + 92129 + 92229 + 92329 + 92429 + 92529 + 92629 + 92729 + 92829 + 92929 + 93039 + 93139 + 93239 + 93339 + 93439 + 93539 + 93639 + 93739 + 93839 + 93939 + 94049$ $+ 94149 + 94249 + 94349 + 94449 + 94549 + 94649 + 94749 + 94849 + 94949 + 95059 + 95159 + 95259 + 95359 + 95459 + 95559 + 95659 + 95759 + 95859 + 95959 + 96069 + 96169 + 96269 + 96369 + 96469 + 96569 + 96669 + 96769 + 96869$ $+ 96969 + 97079 + 97179 + 97279 + 97379 + 97479 + 97579 + 97679 + 97779 + 97879 + 97979 + 98089 + 98189 + 98289 + 98389 + 98489 + 98589 + 98689 + 98789 + 98889 + 98989 + 99099 + 99199 + 99299 + 99399 + 99499 + 99599 + 99699$

\[+ 99799 + 99899 + 99999 = 49500000 = 4 + 9 + 5 = \Boxed{18} \to B\] (Error compiling LaTeX. Unknown error_msg)

.

Solution Three

Notice that 10001+ 99999 = 111000. In fact, ordering the palindromes in ascending order, we find that the sum of the nth palindrome and the nth to last palindrome is 111000. We have 9*10*10 palindromes, or 450 pairs of palindromes summing to 111000. Performing the multiplication gives 49500000, so the sum is 18.

@@ Line 21: / Line 21: @@
 ==Solution Two==
-Hello. As there are only <math>9\cdot10\cdot10 = 900</math> five digit palindromes, it is sufficient to add up all of them.
+As there are only <math>9\cdot10\cdot10 = 900</math> five digit palindromes, it is sufficient to add up all of them.
 <cmath>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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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 + 29492 + 29592 + 29692</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 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</cmath><cmath> + 43734 + 43834 + 43934 + 44044 + 44144 + 44244 + 44344 + 44444 + 44544 + 44644 + 44744 + 44844 + 44944 + 45054 + 45154 + 45254 + 45354 + 45454 + 45554 + 45654 + 45754 + 45854 + 45954 + 46064 + 46164 + 46264 + 46364 + 46464</cmath><cmath> + 46564 + 46664 + 46764 + 46864 + 46964 + 47074 + 47174 + 47274 + 47374 + 47474 + 47574 + 47674 + 47774 + 47874 + 47974 + 48084 + 48184 + 48284 + 48384 + 48484 + 48584 + 48684 + 48784 + 48884 + 48984 + 49094 + 49194 + 49294</cmath><cmath> + 49394 + 49494 + 49594 + 49694 + 49794 + 49894 + 49994 + 50005 + 50105 + 50205 + 50305 + 50405 + 50505 + 50605 + 50705 + 50805 + 50905 + 51015 + 51115 + 51215 + 51315 + 51415 + 51515 + 51615 + 51715 + 51815 + 51915 + 52025</cmath><cmath> + 52125 + 52225 + 52325 + 52425 + 52525 + 52625 + 52725 + 52825 + 52925 + 53035 + 53135 + 53235 + 53335 + 53435 + 53535 + 53635 + 53735 + 53835 + 53935 + 54045 + 54145 + 54245 + 54345 + 54445 + 54545 + 54645 + 54745 + 54845</cmath><cmath> + 54945 + 55055 + 55155 + 55255 + 55355 + 55455 + 55555 + 55655 + 55755 + 55855 + 55955 + 56065 + 56165 + 56265 + 56365 + 56465 + 56565 + 56665 + 56765 + 56865 + 56965 + 57075 + 57175 + 57275 + 57375 + 57475 + 57575 + 57675</cmath><cmath> + 57775 + 57875 + 57975 + 58085 + 58185 + 58285 + 58385 + 58485 + 58585 + 58685 + 58785 + 58885 + 58985 + 59095 + 59195 + 59295 + 59395 + 59495 + 59595 + 59695 + 59795 + 59895 + 59995 + 60006 + 60106 + 60206 + 60306 + 60406</cmath><cmath> + 60506 + 60606 + 60706 + 60806 + 60906 + 61016 + 61116 + 61216 + 61316 + 61416 + 61516 + 61616 + 61716 + 61816 + 61916 + 62026 + 62126 + 62226 + 62326 + 62426 + 62526 + 62626 + 62726 + 62826 + 62926 + 63036 + 63136 + 63236</cmath><cmath> + 63336 + 63436 + 63536 + 63636 + 63736 + 63836 + 63936 + 64046 + 64146 + 64246 + 64346 + 64446 + 64546 + 64646 + 64746 + 64846 + 64946 + 65056 + 65156 + 65256 + 65356 + 65456 + 65556 + 65656 + 65756 + 65856 + 65956 + 66066</cmath><cmath>+ 66166 + 66266 + 66366 + 66466 + 66566 + 66666 + 66766 + 66866 + 66966 + 67076 + 67176 + 67276 + 67376 + 67476 + 67576 + 67676 + 67776 + 67876 + 67976 + 68086 + 68186 + 68286 + 68386 + 68486 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+ 80108 + 80208 + 80308 + 80408 + 80508 + 80608 + 80708 + 80808 + 80908 + 81018 + 81118 + 81218 + 81318 + 81418 + 81518 + 81618 + 81718 + 81818 + 81918 + 82028 + 82128 + 82228 + 82328 + 82428 + 82528 + 82628 + 82728 + 82828</cmath><cmath> + 82928 + 83038 + 83138 + 83238 + 83338 + 83438 + 83538 + 83638 + 83738 + 83838 + 83938 + 84048 + 84148 + 84248 + 84348 + 84448 + 84548 + 84648 + 84748 + 84848 + 84948 + 85058 + 85158 + 85258 + 85358 + 85458 + 85558 + 85658</cmath><cmath> + 85758 + 85858 + 85958 + 86068 + 86168 + 86268 + 86368 + 86468 + 86568 + 86668 + 86768 + 86868 + 86968 + 87078 + 87178 + 87278 + 87378 + 87478 + 87578 + 87678 + 87778 + 87878 + 87978 + 88088 + 88188 + 88288 + 88388 + 88488</cmath><cmath> + 88588 + 88688 + 88788 + 88888 + 88988 + 89098 + 89198 + 89298 + 89398 + 89498 + 89598 + 89698 + 89798 + 89898 + 89998 + 90009 + 90109 + 90209 + 90309 + 90409 + 90509 + 90609 + 90709 + 90809 + 90909 + 91019 + 91119 + 91219</cmath><cmath> + 91319 + 91419 + 91519 + 91619 + 91719 + 91819 + 91919 + 92029 + 92129 + 92229 + 92329 + 92429 + 92529 + 92629 + 92729 + 92829 + 92929 + 93039 + 93139 + 93239 + 93339 + 93439 + 93539 + 93639 + 93739 + 93839 + 93939 + 94049</cmath><cmath> + 94149 + 94249 + 94349 + 94449 + 94549 + 94649 + 94749 + 94849 + 94949 + 95059 + 95159 + 95259 + 95359 + 95459 + 95559 + 95659 + 95759 + 95859 + 95959 + 96069 + 96169 + 96269 + 96369 + 96469 + 96569 + 96669 + 96769 + 96869</cmath><cmath> + 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</cmath><cmath> + 99799 + 99899 + 99999 = 49500000 = 4 + 9 + 5 = \Boxed{18} \to B</cmath>.
 ==Solution Three==
 Notice that 10001+ 99999 = 111000. In fact, ordering the palindromes in ascending order, we find that the sum of the nth palindrome and the nth to last palindrome is 111000. We have 9*10*10 palindromes, or 450 pairs of palindromes summing to 111000. Performing the multiplication gives 49500000, so the sum is 18.

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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