Difference between revisions of "2006 AMC 12A Problems/Problem 2"
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By the definition of <math>\otimes</math>, we have <math>h\otimes h=h^{3}-h</math>. Then <math>h\otimes (h\otimes h)=h\otimes (h^{3}-h)=h^{3}-(h^{3}-h)=h</math>. The answer is <math>\mathrm{(C)}</math>. | By the definition of <math>\otimes</math>, we have <math>h\otimes h=h^{3}-h</math>. Then <math>h\otimes (h\otimes h)=h\otimes (h^{3}-h)=h^{3}-(h^{3}-h)=h</math>. The answer is <math>\mathrm{(C)}</math>. | ||
| + | ==Solution 2== | ||
| + | Substitute <math>1</math> for <math>h</math>. You get <math>1^3-(1^3-1)</math> which is 1. That is <math>h</math>, so the answer is <math>\mathrm{(C)}</math>. | ||
== See also == | == See also == | ||
{{AMC12 box|year=2006|ab=A|num-b=1|num-a=3}} | {{AMC12 box|year=2006|ab=A|num-b=1|num-a=3}} | ||
Revision as of 17:36, 29 April 2021
- The following problem is from both the 2006 AMC 12A #2 and 2006 AMC 10A #2, so both problems redirect to this page.
Contents
Problem
Define 
. What is 
?
Solution
By the definition of 
, we have 
. Then 
. The answer is 
.
Solution 2
Substitute 
for 
. You get 
which is 1. That is , so the answer is .
See also
| 2006 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 1 |
Followed by Problem 3 |
| 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 | |
| 2006 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
