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Difference between revisions of "2021 AIME II Problems/Problem 1"

(→‎Solution 3 (Symmetry and Generalization))
(→‎Solution 6 (Two cases))
 
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==Solution 1==
 
==Solution 1==
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Recall* the the arithmetic mean of all the <math>n</math> digit palindromes is just the average of the largest and smallest <math>n</math> digit palindromes, and in this case the <math>2</math> palindromes are <math>101</math> and <math>999</math> and <math>\frac{101+999}{2}=550</math> and <math>\boxed{550}</math> is the final answer.
+
Recall that the arithmetic mean of all the <math>n</math> digit palindromes is just the average of the largest and smallest <math>n</math> digit palindromes, and in this case the <math>2</math> palindromes are <math>101</math> and <math>999</math> and <math>\frac{101+999}{2}=\boxed{550},</math> which is the final answer.
  
 
~ math31415926535
 
~ math31415926535
−
 
−
* This relies on the fact that whenever x is a palindrome, 1100-x is also a palindrome. Once you realize this bijective relationship you will immediately obtain the mean, although it honestly may not be easy to see it on spot.
 
−
* Refer to <u><b>Solution 3 (Symmetry and Generalization)</b></u> for the note above.
 
−
 
−
-Note by Ross Gao and MRENTHUSIASM
 
  
 
==Solution 2==
 
==Solution 2==
−
For any palindrome <math>\overline{ABA}</math>, note that <math>\overline{ABA}</math>,  is 100A + 10B + A which is also 101A + 10B.  
+
For any palindrome <math>\underline{ABA},</math> note that <math>\underline{ABA}</math> is <math>100A + 10B + A = 101A + 10B.</math>
−
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 = <math>\boxed{550}</math>.
+
The average for <math>A</math> is <math>5</math> since <math>A</math> can be any of <math>1, 2, 3, 4, 5, 6, 7, 8,</math> or <math>9.</math> The average for <math>B</math> is <math>4.5</math> since <math>B</math> is either <math>0, 1, 2, 3, 4, 5, 6, 7, 8,</math> or <math>9.</math> Therefore, the answer is <math>505 + 45 = \boxed{550}.</math>
  
 
- ARCTICTURN
 
- ARCTICTURN
  
 
==Solution 3 (Symmetry and Generalization)==
 
==Solution 3 (Symmetry and Generalization)==
−
For any three-digit palindrome <math>\overline{ABA},</math> where <math>A</math> and <math>B</math> are digits with <math>A\neq0,</math> note that <math>\overline{(10-A)(9-B)(10-A)}</math> must be another palindrome by symmetry. The mapping from 3-digit palindromes to 3-dit palindromes, <math>f: \overline{ABA} \rightarrow \overline{(10-A)(9-B)(10-A)}</math>, is a bijection. Different palindromes are mapped to different palindromes, and each palindrome has a preimage. In particular, because <math>f^2=id</math>, <math>f^{-1}(x)=f(x)</math>. Therefore, we can pair each three-digit palindrome uniquely with another three-digit palindrome so that they sum to  
+
For every three-digit palindrome <math>\underline{ABA}</math> with <math>A\in\{1,2,3,4,5,6,7,8,9\}</math> and <math>B\in\{0,1,2,3,4,5,6,7,8,9\},</math> note that <math>\underline{(10-A)(9-B)(10-A)}</math> 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  
 
<cmath>\begin{align*}
 
<cmath>\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] \\
+
\underline{ABA}+\underline{(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] \\
 
&=\left[100A+10B+A\right]+\left[1000-100A+90-10B+10-A\right] \\
 
&=1000+90+10 \\
 
&=1000+90+10 \\
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For instances:
 
For instances:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
101+999&=1100, \\
+
171+929&=1100, \\
 
262+838&=1100, \\
 
262+838&=1100, \\
−
373+727&=1100, \\
+
303+797&=1100, \\
 
414+686&=1100, \\
 
414+686&=1100, \\
 
545+555&=1100,
 
545+555&=1100,
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From this symmetry, the arithmetic mean of all the three-digit palindromes is <math>\frac{1100}{2}=\boxed{550}.</math>
 
From this symmetry, the arithmetic mean of all the three-digit palindromes is <math>\frac{1100}{2}=\boxed{550}.</math>
 +
 +
<u><b>Remark</b></u>
 +
 +
By the Multiplication Principle, there are <math>9\cdot10=90</math> three-digit palindromes in total. Their sum is <math>1100\cdot45=49500,</math> as we can match them into <math>45</math> pairs such that each pair sums to <math>1100.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
==Solution 4==
+
==Solution 4 (Similar to Solution 2: Very, Very Easy and Quick)==
−
<cmath>\begin{align*}
+
We notice that a three-digit palindrome looks like this: <math>\underline{aba}.</math>
−
\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 \\
+
And we know <math>a</math> can be any digit from <math>1</math> through <math>9,</math> and <math>b</math> can be any digit from <math>0</math> through <math>9,</math> so there are <math>9\times{10}=90</math> three-digit palindromes.
−
&= 101 \cdot 10 \sum_{A = 1}^9 A + 10 \cdot 9 \sum_{B = 0}^9 B \\
+
 
−
&= 1010 \cdot 45 + 90 \cdot 45 \\
+
We want to find the sum of these <math>90</math> palindromes and divide it by <math>90</math> to find the arithmetic mean.
−
&=  
+
 
−
\end{align*}</cmath>
+
How can we do that? Instead of adding the numbers up, we can break each palindrome into two parts: <math>101a+10b.</math>
 +
 
 +
Thus, all of these <math>90</math> palindromes can be broken into this form.
 +
 
 +
Thus, the sum of these <math>90</math> palindromes will be <math>101\times{(1+2+...+9)}\times{10}+10\times{(0+1+2+...+9)}\times{9},</math> because each <math>a</math> will be in <math>10</math> different palindromes (since for each <math>a,</math> there are <math>10</math> choices for <math>b</math>). The same logic explains why we multiply by <math>9</math> when computing the total sum of <math>b.</math>
 +
 
 +
We get a sum of <math>45\times{1100},</math> but don't compute this! Divide this by <math>90</math> and you will get <math>\boxed{550}.</math>
 +
 
 +
~<math>\alpha b \alpha</math>
 +
 
 +
==Solution 5 (Extremely Fast Solution)==
 +
 
 +
The possible values of the first and last digits each are <math>1, 2, ..., 8, 9</math> with a sum of <math>45</math> so the average value is <math>5.</math> The middle digit can be any digit from <math>0</math> to <math>9</math> with a sum of <math>45,</math> so the average value is <math>4.5.</math> The average of all three-digit palindromes is <math>5\cdot 10^2+4.5\cdot 10+5=\boxed{550}.</math>
 +
 
 +
~MathIsFun286
 +
 
 +
~MathFun1000 (Rephrasing with more clarity)
 +
 
 +
==Solution 6 (Two cases)==
 +
<i><b>Case 1</b></i>
 +
 
 +
Consider palindromes of the form <math>5x5 = 505 + 10x.</math> There are <math>10</math> of them. The arithmetic mean of the first term is <math>505,</math> and the second <math>\frac {10 \cdot(0 + 1 + ... + 9)}{10} = 45.</math>
 +
The arithmetic mean of the sum is <math>505 + 45 = 550.</math>
 +
 
 +
<i><b>Case 2</b></i>
 +
 
 +
Consider palindromes of the form <math>yxy,</math> where <math>y= {1,2,3,4,6,7,8,9}.</math>
 +
Let <math>u = 10 – y, v = 9 – x.</math>
 +
Then <math>uvu</math> is a symmetric palindrome that can be constructed for each <math>yxy.</math>
 +
The arithmetic mean of each such pair is <math>550.</math> For example, <math>\frac{737 + 363}{2} = 550.</math>
 +
 
 +
Thus, all palindromes are divided into groups of numbers with the arithmetic mean in each group equal to <math>550.</math>
 +
 
 +
The arithmetic mean of all numbers is also <math>550.</math>
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
  
−
- A bit too complicated of a solution - somebody please fix. - ARCTICTURN
+
==Remark==
 +
Visit the [https://artofproblemsolving.com/wiki/index.php/Talk:2021_AIME_II_Problems/Problem_1 Discussion Page] for questions and further generalizations.
  
−
Doriding is the original author. I will wait for him to come back. ~MRENTHUSIASM
+
~MRENTHUSIASM
  
 
==Video Solution==
 
==Video Solution==
 
https://www.youtube.com/watch?v=jDP2PErthkg
 
https://www.youtube.com/watch?v=jDP2PErthkg
 +
 +
==Video Solution by Interstigation==
 +
https://youtu.be/3_ik5N33CnQ
 +
 +
speedy 2 min video
  
 
==See Also==
 
==See Also==
−
 
 
{{AIME box|year=2021|n=II|before=First Problem|num-a=2}}
 
{{AIME box|year=2021|n=II|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 14:36, 30 August 2022

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 $777$ or $383$.)

Solution 1

Recall that the arithmetic mean of all the $n$ digit palindromes is just the average of the largest and smallest $n$ digit palindromes, and in this case the $2$ palindromes are $101$ and $999$ and $\frac{101+999}{2}=\boxed{550},$ which is the final answer.

~ math31415926535

Solution 2

For any palindrome $\underline{ABA},$ note that $\underline{ABA}$ is $100A + 10B + A = 101A + 10B.$ 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 $505 + 45 = \boxed{550}.$

- ARCTICTURN

Solution 3 (Symmetry and Generalization)

For every three-digit palindrome $\underline{ABA}$ with $A\in\{1,2,3,4,5,6,7,8,9\}$ and $B\in\{0,1,2,3,4,5,6,7,8,9\},$ note that $\underline{(10-A)(9-B)(10-A)}$ 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 \begin{align*} \underline{ABA}+\underline{(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*} For instances: \begin{align*} 171+929&=1100, \\ 262+838&=1100, \\ 303+797&=1100, \\ 414+686&=1100, \\ 545+555&=1100, \end{align*} and so on.

From this symmetry, the arithmetic mean of all the three-digit palindromes is $\frac{1100}{2}=\boxed{550}.$

Remark

By the Multiplication Principle, there are $9\cdot10=90$ three-digit palindromes in total. Their sum is $1100\cdot45=49500,$ as we can match them into $45$ pairs such that each pair sums to $1100.$

~MRENTHUSIASM

Solution 4 (Similar to Solution 2: Very, Very Easy and Quick)

We notice that a three-digit palindrome looks like this: $\underline{aba}.$

And we know $a$ can be any digit from $1$ through $9,$ and $b$ can be any digit from $0$ through $9,$ so there are $9\times{10}=90$ 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 $101\times{(1+2+...+9)}\times{10}+10\times{(0+1+2+...+9)}\times{9},$ because each $a$ will be in $10$ different palindromes (since for each $a,$ there are $10$ choices for $b$). The same logic explains why we multiply by $9$ when computing the total sum of $b.$

We get a sum of $45\times{1100},$ but don't compute this! Divide this by $90$ and you will get $\boxed{550}.$

~$\alpha b \alpha$

Solution 5 (Extremely Fast Solution)

The possible values of the first and last digits each are $1, 2, ..., 8, 9$ with a sum of $45$ so the average value is $5.$ The middle digit can be any digit from $0$ to $9$ with a sum of $45,$ so the average value is $4.5.$ The average of all three-digit palindromes is $5\cdot 10^2+4.5\cdot 10+5=\boxed{550}.$

~MathIsFun286

~MathFun1000 (Rephrasing with more clarity)

Solution 6 (Two cases)

Case 1

Consider palindromes of the form $5x5 = 505 + 10x.$ There are $10$ of them. The arithmetic mean of the first term is $505,$ and the second $\frac {10 \cdot(0 + 1 + ... + 9)}{10} = 45.$ The arithmetic mean of the sum is $505 + 45 = 550.$

Case 2

Consider palindromes of the form $yxy,$ where $y= {1,2,3,4,6,7,8,9}.$ Let $u = 10 – y, v = 9 – x.$ Then $uvu$ is a symmetric palindrome that can be constructed for each $yxy.$ The arithmetic mean of each such pair is $550.$ For example, $\frac{737 + 363}{2} = 550.$

Thus, all palindromes are divided into groups of numbers with the arithmetic mean in each group equal to $550.$

The arithmetic mean of all numbers is also $550.$

vladimir.shelomovskii@gmail.com, vvsss

Remark

Visit the Discussion Page for questions and further generalizations.

~MRENTHUSIASM

Video Solution

https://www.youtube.com/watch?v=jDP2PErthkg

Video Solution by Interstigation

https://youtu.be/3_ik5N33CnQ

speedy 2 min video

See Also

2021 AIME II (Problems • Answer Key • Resources)
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First Problem 		Followed by
Problem 2
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