Difference between revisions of "2019 AMC 8 Problems/Problem 13"

Revision as of 07:38, 20 January 2025

Problem

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 $N$ ?

$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

Solution 1

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 $1+1+0 = \boxed{\textbf{(A) }2}$ .

There are other sets of 2-digit numbers that satisfy this rule. Some of them are $110 = 11+33+66$ and $110 = 22+33+55$

Video Solution by EZLX

https://youtu.be/K_3NHSUo7mw

~COOKEMONSTER SIRCOOKIES EZLX

Geez EzLx’s work is genius. Amazing

Video Solution 4

https://youtu.be/WeQuJEQKVdo

~savannahsolver

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 18: / Line 18: @@
 Geez EzLx’s work is genius.
 Amazing
-== Video Solution 3==
-Solution detailing how to solve the problem: https://www.youtube.com/watch?v=PJpDJ23sOJM&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=14
 ==Video Solution 4==

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