Difference between revisions of "2006 AMC 12A Problems/Problem 2"
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+ | {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #2]] and [[2006 AMC 10A Problems/Problem 2|2006 AMC 10A #2]]}} | ||
== Problem == | == Problem == | ||
Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>? | Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>? | ||
− | <math>\ | + | <math>\textbf{(A)}\ -h\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ h\qquad\textbf{(D)}\ 2h\qquad\textbf{(E)}\ h^3</math> |
− | == Solution == | + | == Solution 1 == |
− | 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)= | + | 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)=\boxed{\textbf{(C) }h}.</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}} | ||
+ | {{AMC10 box|year=2006|ab=A|num-b=1|num-a=3}} | ||
+ | {{MAA Notice}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Latest revision as of 14:03, 27 July 2023
- The following problem is from both the 2006 AMC 12A #2 and 2006 AMC 10A #2, so both problems redirect to this page.
Problem
Define . What is
?
Solution 1
By the definition of , we have
.
Then,
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.