Problem:
A palindrome is a number that has the same value when read from left to right or from right to left. (For example, is a palindrome.) Let be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
: The two-digit palindromes are , all of which are multiples of . A sum of three of these palindromes will also be a multiple of . The smallest three-digit multiple of is , and 110 can be expressed as the sum of three two-digit palindromes. For example, . The sum of the digits of is .
Note: Javier Cilleruelo, Florian Luca, and Lewis Baxter recently proved that any positive inte-ger can be written as the sum of three palindromes. (See https://arxiv.org/pdf/1602.06208.pdf.)
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions