Difference between revisions of "2003 AMC 12A Problems/Problem 6"
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+ | {{duplicate|[[2003 AMC 12A Problems|2003 AMC 12A #6]] and [[2003 AMC 10A Problems|2003 AMC 10A #6]]}} | ||
== Problem == | == Problem == | ||
Define <math>x \heartsuit y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. Which of the following statements is not true? | Define <math>x \heartsuit y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. Which of the following statements is not true? | ||
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<math>|x| \neq x</math> when <math>x<0</math>, but statement C says that it does for all <math>x</math>. | <math>|x| \neq x</math> when <math>x<0</math>, but statement C says that it does for all <math>x</math>. | ||
− | Therefore the statement that is not true is | + | Therefore the statement that is not true is <math>\boxed{\mathrm{(C)}\ x\heartsuit 0=x\ \text{for all}\ x}</math> |
== See Also == | == See Also == | ||
+ | {{AMC10 box|year=2003|ab=A|num-b=5|num-a=7}} | ||
{{AMC12 box|year=2003|ab=A|num-b=5|num-a=7}} | {{AMC12 box|year=2003|ab=A|num-b=5|num-a=7}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 13:44, 30 July 2011
- The following problem is from both the 2003 AMC 12A #6 and 2003 AMC 10A #6, so both problems redirect to this page.
Problem
Define to be for all real numbers and . Which of the following statements is not true?
for all and
for all and
for all
for all
if
Solution
Examining statement C:
when , but statement C says that it does for all .
Therefore the statement that is not true is
See Also
2003 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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 |
2003 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 |