Difference between revisions of "2003 AMC 10A Problems/Problem 6"
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Therefore the statement that is not true is "<math>x \heartsuit 0 = x</math> for all <math>x</math>" <math>\Rightarrow C</math> | Therefore the statement that is not true is "<math>x \heartsuit 0 = x</math> for all <math>x</math>" <math>\Rightarrow C</math> | ||
− | Alternatively, consider that the given "heart function" is actually the definition of the distance between two points. Examining all of the statements, only C is not necessarily true; if c is negative, the distance between c and 0 is the absolute value of c, not c itself, because distance is always nonnegative. | + | Alternatively, consider that the given "heart function" is actually the definition of the distance between two points. Examining all of the statements, only C is not necessarily true; if c is negative, the distance between <math>c</math> and <math>0</math> is the absolute value of <math>c</math>, not <math>c</math> itself, because distance is always nonnegative. |
== See Also == | == See Also == |
Revision as of 20:55, 19 May 2008
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 D says that it does for all .
Therefore the statement that is not true is " for all "
Alternatively, consider that the given "heart function" is actually the definition of the distance between two points. Examining all of the statements, only C is not necessarily true; if c is negative, the distance between and is the absolute value of , not itself, because distance is always nonnegative.
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 |