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. 
.
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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