2019 AMC 8 Problems/Problem 13
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Problem 13
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 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 ?
Solution 1
Note that the only positive 2-digit palindromes are multiples of 11, namely . Since is the sum of 2-digit palindromes, is necessarily a multiple of 11. The smallest 3-digit multiple of 11 which is not a palindrome is 110, so is a candidate solution. We must check that 110 can be written as the sum of three distinct 2-digit palindromes; this suffices as . Then, , and the sum of the digits of is .
- There are other sets of 2-digit numbers that satisfy this rule. Some of them are and
Solution 2 (variant of Solution 1)
We already know that two-digit palindromes can only be two-digit multiples of 11; which are: and . Since this is clear, we will need to find out the least multiple of 11 that is not a palindrome. Then, we start counting. Aha! This multiple of 11, 110, not only isn’t a palindrome, but it also is the sum of three distinct two-digit palindromes, for example: 11 + 22 + 77, 22 + 33 + 55, and 44 + 11 + 55! The sum of ’s digits is .
Thank you to the writer of Solution 1 for inspiring me to create this!
EarthSaver 15:13, 11 June 2021 (EDT)
Solution 3 (basically a version of the above solutions)
As stated above, two-digit palindromes can only be two-digit multiples of 11. We can see that if we add anything that are multiples of 11 together, we will again get a multiple of 11. For instance, . Since we know this fact and we are finding the smallest value possible, we can start with the first three-digit multiple of 11 which is . Since this is not a palindrome and can be the sum of 3 two-digit palindromes (see above solutions for more details), fits the bill. We can see that the sum of 's digits is .
~yeye
Video Solution 1
https://youtu.be/gOZOCFNXMhE ~ The Learning Royal
Video Solution 2
https://www.youtube.com/watch?v=bOnNFeZs7S8
Video Solution 3
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=PJpDJ23sOJM&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=14
Video Solution 4
~savannahsolver
See also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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