2010 AMC 10A Problem 9

Question

A palindrome, such as

83438,

is a number that remains the same when its digits are reversed. The numbers

x

and

x + 32

are three-digit and four-digit palindromes, respectively. What is the sum of the digits of

x?

20

21

22

23

24

Bound the four-digit palindrome between

1000

and

1031

Find the only palindrome in that interval, then subtract

32

Note that

x

is at most

999.

This means that

x + 32

has a maximum of

1031.

Similarly, we have that the minimum value of

x + 32

is

1000.

The only palindrome in this range is

1001,

so this is what

x + 32

equals.

Then

x + 32 = 1001

x = 969.

The sum of the digits is then

9 + 6 + 9 = 24.

Thus, E is the correct answer.

Problem 9 in other years