Difference between revisions of "2021 Fall AMC 12B Problems/Problem 7"
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<math>\textbf{(E)} \: x+y+z=1</math> | <math>\textbf{(E)} \: x+y+z=1</math> | ||
+ | ==Solution 1== | ||
+ | Plugging in every choice, we see that choice <math>\textbf{(D)}</math> works. | ||
+ | |||
+ | |||
+ | We have <math>y=x+1, z=x</math>, so | ||
+ | <cmath>x(x-y)+y(y-z)+z(z-x)=x(x-(x+1))+(x+1)((x+1)-x)+x(x-x)=x(-1)+(x+1)(1)=1.</cmath> | ||
+ | Our answer is <math>\textbf{(D)}</math>. | ||
+ | |||
+ | ~kingofpineapplz | ||
− | ==Solution | + | ==Solution 2 (Bash) == |
Just plug in all these options one by one, and one sees that all but <math>D</math> fails to satisfy the equation. | Just plug in all these options one by one, and one sees that all but <math>D</math> fails to satisfy the equation. | ||
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~Wilhelm Z | ~Wilhelm Z | ||
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{{AMC12 box|year=2021 Fall|ab=B|num-a=8|num-b=6}} | {{AMC12 box|year=2021 Fall|ab=B|num-a=8|num-b=6}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 10:34, 24 November 2021
- The following problem is from both the 2021 Fall AMC 10B #12 and 2021 Fall AMC 12B #7, so both problems redirect to this page.
Problem
Which of the following conditions is sufficient to guarantee that integers , , and satisfy the equation
and
and
and
and
Solution 1
Plugging in every choice, we see that choice works.
We have , so
Our answer is .
~kingofpineapplz
Solution 2 (Bash)
Just plug in all these options one by one, and one sees that all but fails to satisfy the equation.
For , substitute and :
Hence the answer is
~Wilhelm Z
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.