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? | ||
| Line 13: | Line 14: | ||
== Solution == | == Solution == | ||
| − | Examining statement C: | + | We start by looking at the answers. Examining statement C, we notice: |
<math> x \heartsuit 0 = |x-0| = |x| </math> | <math> x \heartsuit 0 = |x-0| = |x| </math> | ||
| Line 19: | Line 20: | ||
<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> |
| + | |||
| + | == Video Solution == | ||
| + | |||
| + | https://www.youtube.com/watch?v=d-IRsaIUdDA ~David | ||
== 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]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 15:38, 19 August 2023
- The following problem is from both the 2003 AMC 12A #6 and 2003 AMC 10A #6, so both problems redirect to this page.
Contents
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
We start by looking at the answers. Examining statement C, we notice:
when 
, but statement C says that it does for all .
Therefore the statement that is not true is 
Video Solution
https://www.youtube.com/watch?v=d-IRsaIUdDA ~David
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
