Difference between revisions of "2002 AMC 12P Problems/Problem 20"

m (Solution)
(Solution)
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
{{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 {\text{(B) }2000}</math>.
+
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 $f$ be a real-valued function such that

\[f(x) + 2f(\frac{2002}{x}) = 3x\]

for all $x>0.$ Find $f(2).$

$\text{(A) }1000 \qquad \text{(B) }2000 \qquad \text{(C) }3000 \qquad \text{(D) }4000 \qquad \text{(E) }6000$

Solution

Setting $x = 2$ gives $f(2) + 2f(1001) = 6$. Setting $x = 1001$ gives $2f(2) + f(1001) = 3003$.

Adding these 2 equations and dividing by 3 gives $f(2) + f(1001) = \frac{6+3003}{3} = 1003$.

Subtracting these 2 equations gives $f(2) - f(1001) = 2997$.

Therefore, $f(2) = \frac{1003+2997}{2} = \boxed {\textbf{(B) }2000}$.

See also

2002 AMC 10P (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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. AMC logo.png