2014 AMC 12A Problems/Problem 15
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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
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 so the sum of the digits of the sum is
, or
.
(Solution by AwesomeToad)