2002 AMC 12P Problems/Problem 20

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 $f(2).$

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

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

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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All AMC 12 Problems and Solutions