2021 AIME II Problems/Problem 1: Difference between revisions
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The number takes the form <math>\overline{ABA}</math>, where <math>A \in \left\{ 1 , 2 , \cdots , 9 \right\}</math> and <math>B \in \left\{ 0 , 1 , \cdots , 9 \right\}</math>. | The number takes the form <math>\overline{ABA}</math>, where <math>A \in \left\{ 1 , 2 , \cdots , 9 \right\}</math> and <math>B \in \left\{ 0 , 1 , \cdots , 9 \right\}</math>. | ||
Therefore, the sum of all such numbers is equal to | |||
\begin{align*} | |||
\sum_{A = 1}^9 \sum_{B = 0}^9 \overline{ABA} | |||
& = \sum_{A = 1}^9 \sum_{B = 0}^9 \left( 101 A + 10 B \right) \\ | |||
& = \sum_{A = 1}^9 \sum_{B = 0}^9 101 A + \sum_{A = 1}^9 \sum_{B = 0}^9 10 B \\ | |||
& = 101 \cdot 10 \sum_{A = 1}^9 A + 10 \cdot 9 \sum_{B = 0}^9 B \\ | |||
& = 1010 \cdot 45 + 90 \cdot 45 \\ | |||
& = | |||
\end{align*} | |||
~ Steven Chen | ~ Steven Chen | ||
Revision as of 16:17, 22 March 2021
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)
For any three-digit palindrome 
where 
and 
are digits and 
note that 
must be another palindrome by symmetry. This means we can pair each three-digit palindrome uniquely with another three-digit palindrome so that they sum to
![\begin{align*} \overline{ABA}+\overline{(10-A)(9-B)(10-A)}&=\left[100A+10B+A\right]+\left[100(10-A)+10(9-B)+(10-A)\right] \\ &=\left[100A+10B+A\right]+\left[1000-100A+90-10B+10-A\right] \\ &=1000+90+10 \\ &=1100. \end{align*}](http://latex.artofproblemsolving.com/f/d/6/fd6c776254e3f02e57f957c964234ba0e2cd6722.png)
For instances:
and so on.
From this symmetry, the arithmetic mean of all the three-digit palindromes is 
~MRENTHUSIASM
Solution 4
The number takes the form , where 
and 
.
Therefore, the sum of all such numbers is equal to
\begin{align*} \sum_{A = 1}^9 \sum_{B = 0}^9 \overline{ABA} & = \sum_{A = 1}^9 \sum_{B = 0}^9 \left( 101 A + 10 B \right) \\ & = \sum_{A = 1}^9 \sum_{B = 0}^9 101 A + \sum_{A = 1}^9 \sum_{B = 0}^9 10 B \\ & = 101 \cdot 10 \sum_{A = 1}^9 A + 10 \cdot 9 \sum_{B = 0}^9 B \\ & = 1010 \cdot 45 + 90 \cdot 45 \\ & = \end{align*}
~ Steven Chen
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