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

Revision as of 00:01, 30 December 2023

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

If $\log_{b} 729 = n$ , then $b^n = 729$ . Since $729 = 3^6$ , $b$ must be $3$ to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of $b \Longrightarrow \boxed{\mathrm{E}}$ .

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.

@@ Line 2: / Line 2: @@
 Let <math>f</math> be a real-valued function such that
 <cmath>f(x) + 2f(\frac{2002}{x}) =3x</cmath>
+for all <math>x>0.</math> Find <math>f(2).</math>
 <math>
 \text{(A) }1000

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

Revision as of 00:01, 30 December 2023

Problem

Solution

See also