2021 AIME II Problems/Problem 1
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 .)
Recall that 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 which is the final answer.
For any palindrome note that is The average for is since can be any of or The average for is since is either or Therefore, the answer is
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
By the Multiplication Principle, there are three-digit palindromes in total. Their sum is as we can match them into pairs such that each pair sums to
Solution 4 (Similar to Solution 2: Very, Very Easy and Quick)
We notice that a three-digit palindrome looks like this:
And we know can be any digit from through and can be any digit from through so there are three-digit palindromes.
We want to find the sum of these palindromes and divide it by to find the arithmetic mean.
How can we do that? Instead of adding the numbers up, we can break each palindrome into two parts:
Thus, all of these palindromes can be broken into this form.
Thus, the sum of these palindromes will be because each will be in different palindromes (since for each there are choices for ). The same logic explains why we multiply by when computing the total sum of
We get a sum of but don't compute this! Divide this by and you will get
Solution 5 (Extremely Fast Solution)
The possible values of the first and last digits each are with a sum of so the average value is The middle digit can be any digit from to with a sum of so the average value is The average of all three-digit palindromes is
~MathFun1000 (Rephrasing with more clarity)
Solution 6 (Two cases)
Consider palindromes of the form There are of them. The arithmetic mean of the first term is and the second The arithmetic mean of the sum is
Consider palindromes of the form where Let Then is a symmetric palindrome that can be constructed for each The arithmetic mean of each such pair is For example,
Thus, all palindromes are divided into groups of numbers with the arithmetic mean in each group equal to
The arithmetic mean of all numbers is also
Visit the Discussion Page for questions and further generalizations.
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