Talk:2021 AIME II Problems/Problem 1
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)
The arithmetic mean of all values for is
and the arithmetic mean of all values for each of
is
Together, the arithmetic mean of all the
-digit palindromes is
~MRENTHUSIASM
Proof 2 (Generalization of Solution 3)
Note that must be another palindrome by symmetry. Therefore, we can pair each
-digit palindrome
uniquely with another
-digit palindrome
so that they sum to
From this symmetry, the arithmetic mean of all the
-digit palindromes is
As a side note, the total number of -digit palindromes is
by the Multiplication Principle. Their sum is
as we can match them into
pairs such that each pair sums to
~MRENTHUSIASM