Difference between revisions of "2020 AMC 8 Problems/Problem 22"

(Solution 2)
(Solution 2)
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The last set in this sequence contains all the numbers which will lead to the number 1 after the 6-step process is repeated. The sum of these numbers is <math>1+8+10+64=83\implies\boxed{\textbf{(E) }83}</math>.<br>
 
The last set in this sequence contains all the numbers which will lead to the number 1 after the 6-step process is repeated. The sum of these numbers is <math>1+8+10+64=83\implies\boxed{\textbf{(E) }83}</math>.<br>
 
~ junaidmansuri
 
~ junaidmansuri
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==Solution 3==
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The most straightforward solutions is just working backwards with a diagram as shown below:
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<img>https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC8yLzc0MjY0Yjc2NDcyMDgwNTBlZDlhMDlmNjNmMjI1Y2RhNmUwODkyLnBuZw==&rn=MjAyMCBBTUMgOCBTb2x1dGlvbiAyMi5wbmc=</img>
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Hence, the answer is <math>1+8+64+10=\textbf{(E) }83</math>.
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-franzliszt
  
 
==See also==  
 
==See also==  
 
{{AMC8 box|year=2020|num-b=21|num-a=23}}
 
{{AMC8 box|year=2020|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:08, 18 November 2020

Problem 22

When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy] For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\[7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\[N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1\] $\textbf{(A) }73 \qquad \textbf{(B) }74 \qquad \textbf{(C) }75 \qquad \textbf{(D) }82 \qquad \textbf{(E) }83$

Solution 1

We see that $1,8,10,64$ work, so $\boxed{\textbf{(E) }83}$. ~yofro

Solution 2

Start with the final output which is $1$ and then work backwards, carefully including all the possible inputs that could have resulted in that output. For example, for the number $2$, if you go backwards, you only get to $4$, because $4$ is the only input which can lead to an output of $2$. However, for a number like $16$ for example, both the inputs $5$ and $32$ lead to an output of $16$. A nice way to draw this out is to make a tree diagram but one can also make a series of sets which contain all the possible inputs up to that point.

$\{1\}\leftarrow\{2\}\leftarrow\{4\}\leftarrow\{1,8\}\leftarrow\{2,16\}\leftarrow\{4,5,32\}\leftarrow\{1,8,10,64\}$

The last set in this sequence contains all the numbers which will lead to the number 1 after the 6-step process is repeated. The sum of these numbers is $1+8+10+64=83\implies\boxed{\textbf{(E) }83}$.
~ junaidmansuri

Solution 3

The most straightforward solutions is just working backwards with a diagram as shown below:

<img>https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC8yLzc0MjY0Yjc2NDcyMDgwNTBlZDlhMDlmNjNmMjI1Y2RhNmUwODkyLnBuZw==&rn=MjAyMCBBTUMgOCBTb2x1dGlvbiAyMi5wbmc=</img>


Hence, the answer is $1+8+64+10=\textbf{(E) }83$.

-franzliszt

See also

2020 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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All AJHSME/AMC 8 Problems and Solutions

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