2014 AMC 12A Problem 15

Question

A five-digit palindrome is a positive integer with respective digits

abcba,

where

a

is not zero. Let

S

be the sum of all five-digit palindromes. What is the sum of the digits of

S?

9

18

27

36

45

A palindrome

\overline{abcba}

equals

10001a+1010b+100c

Each

a

appears in

10\cdot10

palindromes, each

b

in

9\cdot10

, each

c

in

9\cdot10

; combine with the digit sum

45

Write

\overline{abcba}=10001a+1010b+100c.

Summing over all palindromes, each value of

a\in\{1,\dots,9\}

occurs with

10\cdot10=100

choices of

b,c,

and each value of

b

or

c

occurs with

9\cdot10=90

choices of the other two digits.

Using

\sum a=\sum b=\sum c=45,

\begin{gathered} \scriptsize S=45\big(10001\cdot100+1010\cdot90+100\cdot90\big)\\ =45\cdot1{,}100{,}000\\ =49{,}500{,}000. \end{gathered}

The sum of the digits of

S

is

4+9+5=18.

Thus, the correct answer is B.

Problem 15 in other years