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

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== Problem ==
 
== Problem ==
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How many positive [[integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer?
+
Let <math>f</math> be a real-valued function such that
−
 
+
<cmath>f(x) + 2f(\frac{2002}{x}) =3x</cmath>
−
<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math>
+
<math>
 +
\text{(A) }1000
 +
\qquad
 +
\text{(B) }2000
 +
\qquad
 +
\text{(C) }3000
 +
\qquad
 +
\text{(D) }4000
 +
\qquad
 +
\text{(E) }6000
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</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 23:53, 29 December 2023

Problem

Let $f$ be a real-valued function such that \[f(x) + 2f(\frac{2002}{x}) =3x\] $\text{(A) }1000 \qquad \text{(B) }2000 \qquad \text{(C) }3000 \qquad \text{(D) }4000 \qquad \text{(E) }6000$

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 $b \Longrightarrow \boxed{\mathrm{E}}$.

See also

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

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