Difference between revisions of "2002 AMC 12P Problems/Problem 20"
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+ | {{duplicate|[[2002 AMC 12P Problems|2002 AMC 12P #20]] and [[2002 AMC 10P Problems|2002 AMC 10P #21]]}} | ||
+ | |||
== Problem == | == Problem == | ||
Let <math>f</math> be a real-valued function such that | Let <math>f</math> be a real-valued function such that | ||
Line 28: | Line 30: | ||
<math>f(2) - f(1001) = 2997</math>. | <math>f(2) - f(1001) = 2997</math>. | ||
− | Therefore, <math>f(2) = \frac{1003+2997}{2} = \boxed {\ | + | Therefore, <math>f(2) = \frac{1003+2997}{2} = \boxed {\textbf{(B) }2000}</math>. |
== See also == | == See also == | ||
+ | {{AMC10 box|year=2002|ab=P|num-b=20|num-a=22}} | ||
{{AMC12 box|year=2002|ab=P|num-b=19|num-a=21}} | {{AMC12 box|year=2002|ab=P|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 16:44, 15 July 2024
- The following problem is from both the 2002 AMC 12P #20 and 2002 AMC 10P #21, so both problems redirect to this page.
Problem
Let be a real-valued function such that
for all Find
Solution
Setting gives . Setting gives .
Adding these 2 equations and dividing by 3 gives .
Subtracting these 2 equations gives .
Therefore, .
See also
2002 AMC 10P (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 |
2002 AMC 12P (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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.