Difference between revisions of "Talk:2021 AIME II Problems/Problem 1"
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From this symmetry, the arithmetic mean of all the <math>(2k-1)</math>-digit palindromes is <math>\frac{10^{2k-1}+10^{2k-2}}{2}.</math> | From this symmetry, the arithmetic mean of all the <math>(2k-1)</math>-digit palindromes is <math>\frac{10^{2k-1}+10^{2k-2}}{2}.</math> | ||
− | As a side note, the total number of <math>(2k-1)</math>-digit palindromes is <math>9\cdot10^{k-1}</math> by the Multiplication Principle. Their sum is <math>\left(10^{2k-1}+10^{2k-2}\right)\cdot\frac{9\cdot10^{k-1}}{2}=\frac{9\cdot\left(10^{3k-2}+10^{3k-3}\right)}{2},</math> as we can match them into <math>\frac{9\cdot10^{k-1}}{2}</math> pairs such that | + | As a side note, the total number of <math>(2k-1)</math>-digit palindromes is <math>9\cdot10^{k-1}</math> by the Multiplication Principle. Their sum is <math>\left(10^{2k-1}+10^{2k-2}\right)\cdot\frac{9\cdot10^{k-1}}{2}=\frac{9\cdot\left(10^{3k-2}+10^{3k-3}\right)}{2},</math> as we can match them into <math>\frac{9\cdot10^{k-1}}{2}</math> pairs such that each pair sums to <math>10^{2k-1}+10^{2k-2}.</math> |
~MRENTHUSIASM | ~MRENTHUSIASM |
Revision as of 19:14, 18 July 2021
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