Difference between revisions of "2020 AMC 8 Problems/Problem 22"
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==Solution 2== | ==Solution 2== | ||
− | Start with the final output which is <math>1</math> and then work backwards, carefully including all the possible inputs that could have resulted in that output. For example, for the number <math>2</math>, if you go backwards, you only get to <math>4</math>, because <math>4</math> is the only input which can lead to an output of <math>2</math>. However, for a number like <math>16</math> for example, both the inputs <math>5</math> and <math>32</math> lead to an output of <math>16</math>. 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.<br> | + | Start with the final output which is <math>1</math> and then work backwards, carefully including all the possible inputs that could have resulted in that output. For example, for the number <math>2</math>, if you go backwards, you only get to <math>4</math>, because <math>4</math> is the only input which can lead to an output of <math>2</math>. However, for a number like <math>16</math> for example, both the inputs <math>5</math> and <math>32</math> lead to an output of <math>16</math>. 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.<br><br> |
− | <math>{1}\leftarrow{2}\leftarrow{4}\leftarrow{1,8}\leftarrow{2,16}\leftarrow{4,5,32}\leftarrow{1,8,10,64}</math><br> | + | <math>{1}\leftarrow{2}\leftarrow{4}\leftarrow{1,8}\leftarrow{2,16}\leftarrow{4,5,32}\leftarrow{1,8,10,64}</math><br><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> | 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 |
Revision as of 07:52, 18 November 2020
Contents
Problem 22
When a positive integer is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of the machine will output Then if the output is repeatedly inserted into the machine five more times, the final output is When the same -step process is applied to a different starting value of the final output is What is the sum of all such integers
Solution 1
We see that work, so . ~yofro
Solution 2
Start with the final output which is and then work backwards, carefully including all the possible inputs that could have resulted in that output. For example, for the number , if you go backwards, you only get to , because is the only input which can lead to an output of . However, for a number like for example, both the inputs and lead to an output of . 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.
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 .
~ junaidmansuri
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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