2021 AIME II Problems/Problem 1
Contents
Problem
Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as or .)
Solution 1
Recall the the arithmetic mean of all the digit palindromes is just the average of the largest and smallest digit palindromes, and in this case the palindromes are and and and is the final answer.
~ math31415926535
Solution 2
For any palindrome , note that is 100A + 10B + A, which is also 101A + 10B. The average for A is 5 since A can be any of 1, 2, 3, 4, 5, 6, 7, 8, or 9. The average for B is 4.5 since B is either 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Therefore, the answer is 505 + 45 = .
- ARCTICTURN
Solution 3 (Symmetry and Generalization)
For every three-digit palindrome with and note that must be another palindrome by symmetry. Therefore, we can pair each three-digit palindrome uniquely with another three-digit palindrome so that they sum to For instances: and so on.
From this symmetry, the arithmetic mean of all the three-digit palindromes is
Remark
By the Multiplication Principle, there are three-digit palindromes in total. We can match them into pairs such that the sum of each pair is Therefore, the arithmetic mean of all the three-digit palindromes is
~MRENTHUSIASM
Solution 4 (Very, Very Easy and Quick)
We notice that a three-digit palindrome looks like this
And we know a can be any number from 1-9, and b can be any number from 0-9, so there are three-digit palindromes
We want to find the sum of these 90 palindromes and divide it by 90 to find the arithmetic mean
How can we do that? Instead of adding the numbers up, we can break each palindrome into two parts: 101a+10b
Thus, all of these 90 palindromes can be broken into this form
Thus, the sum of these 90 palindromes will be , because each a will be in 10 different palindromes (since for each a, there are 10 choices for b). The same logic explains why there is a times 9 when computing the total sum of b.
We get a sum of
But don't compute this! There's no need. Divide this by 90 and you will get
~
Solution 5 (Extremely Fast Solution)
The average values of the first and last digits are each and the average value of the middle digit is so the average of all three-digit palindromes is
~MathIsFun286
Video Solution
https://www.youtube.com/watch?v=jDP2PErthkg
Remarks (Further Generalizations)
More generally, for every positive integer the arithmetic mean of all the -digit palindromes is In this problem we have from which the answer is
Note that all -digit palindromes are of the form where and Using this notation, we will prove the bolded claim in two different ways:
Proof 1 (Generalization of Solution 2)
Proof in progress. Please do not edit--a million thanks.
~MRENTHUSIASM
Proof 2 (Generalization of Solution 3)
Proof in progress. Please do not edit--a million thanks.
~MRENTHUSIASM
See Also
2021 AIME II (Problems • Answer Key • Resources) | ||
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