2014 AMC 12A Problems/Problem 15
Contents
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 One
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 ![\[(1+2+\cdots+9)(100)(10^4+1)+(1+2+\cdots+9)(90)(10^3+10)+(1+2+\cdots+9)(90)(10^2)=49500000\]](http://latex.artofproblemsolving.com/3/8/b/38b77fb7c65e53c50aeee86e0951039d75743e95.png)
so the sum of the digits of the sum is 
, or 
.
(Solution by AwesomeToad)
Solution Two
As there are only 
five digit palindromes, it is sufficient to add up all of them. 
+ 
+ 
$78387 + 78487 + 78587 + 78687 + 78787 + 78887 + 78987 + 79097 + 79197 + 79297 + 79397 + 79497 + 79597 + 79697 + 79797 + 79897 + 79997 + 80008 + 80108 + 80208 + 80308 + 80408 + 80508 + 80608 + 80708 + 80808 + 80908 + 81018 + 81118 + 81218 + 81318 + 81418 + 81518 + 81618 + 81718 + 81818 + 81918 + 82028 + 82128 + 82228 + 82328 + 82428 + 82528 + 82628 + 82728 + 82828 + 82928 + 83038 + 83138 + 83238 + 83338 + 83438 + 83538 + 83638 + 83738 + 83838 + 83938 + 84048 + 84148 + 84248 + 84348 + 84448 + 84548 + 84648 + 84748 + 84848 + 84948 + 85058 + 85158 + 85258 + 85358 + 85458 + 85558 + 85658 + 85758 + 85858 + 85958 + 86068 + 86168 + 86268 + 86368 + 86468 + 86568 + 86668 + 86768 + 86868 + 86968 + 87078 + 87178 + 87278 + 87378 + 87478 + 87578 + 87678 + 87778 + 87878 + 87978 + 88088 + 88188 + 88288 + 88388 + 88488 + 88588 + 88688 + 88788 + 88888 + 88988 + 89098 + 89198 + 89298 + 89398 + 89498 + 89598 + 89698 + 89798 + 89898 + 89998 + 90009 + 90109 + 90209 + 90309 + 90409 + 90509 + 90609 + 90709 + 90809 + 90909 + 91019 + 91119 + 91219 + 91319 + 91419 + 91519 + 91619 + 91719 + 91819 + 91919 + 92029 + 92129 + 92229 + 92329 + 92429 + 92529 + 92629 + 92729 + 92829 + 92929 + 93039 + 93139 + 93239 + 93339 + 93439 + 93539 + 93639 + 93739 + 93839 + 93939 + 94049 + 94149 + 94249 + 94349 + 94449 + 94549 + 94649 + 94749 + 94849 + 94949 + 95059 + 95159 + 95259 + 95359 + 95459 + 95559 + 95659 + 95759 + 95859 + 95959 + 96069 + 96169 + 96269 + 96369 + 96469 + 96569 + 96669 + 96769 + 96869 + 96969 + 97079 + 97179 + 97279 + 97379 + 97479 + 97579 + 97679 + 97779 + 97879 + 97979 + 98089 + 98189 + 98289 + 98389 + 98489 + 98589 + 98689 + 98789 + 98889 + 98989 + 99099 + 99199 + 99299 + 99399 + 99499 + 99599 + 99699 + 99799 + 99899 + 99999 = 49500000 = 4 + 9 + 5 = \Boxed{18} \Rightarrow B$ (Error compiling LaTeX. Unknown error_msg).
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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