2019 AMC 8

Complete problem set with solutions and individual problem pages

Problem 13 Medium

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let N 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 N?

  • A.

    2

  • B.

    3

  • C.

    4

  • D.

    5

  • E.

    6

Answer:A

Note that the only positive 2-digit palindromes are multiples of 11, namely 11, 22, \ldots, 99. Since N is the sum of 2-digit palindromes, N is necessarily a multiple of 11. The smallest 3-digit multiple of 11 which is not a palindrome is 110, so N=110 is a candidate solution. We must check that 110 can be written as the sum of three distinct 2-digit palindromes; this suffices as 110=77+22+11. Then, N = 110, and the sum of the digits of N is 1+1+0 = \boxed{\textbf{(A) }2}.

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