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Difference between revisions of "2004 AMC 12A Problems/Problem 17"

(→‎Solution 1 (Forwards))
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What is the value of <math>f(2^{100})</math>?
 
What is the value of <math>f(2^{100})</math>?
  
−
<math>\text {(A)}\ 1 \qquad \text {(B)}\ 2^{99} \qquad \text {(C)}\ 2^{100} \qquad \text {(D)}\ 2^{4950} \qquad \text {(E)}\ 2^{9999}</math>
+
<math>\textbf {(A)}\ 1 \qquad \textbf {(B)}\ 2^{99} \qquad \textbf {(C)}\ 2^{100} \qquad \textbf {(D)}\ 2^{4950} \qquad \textbf {(E)}\ 2^{9999}</math>
  
−
== Solution 1 (Forwards) ==
+
== Solution 1 (Forward) ==
−
Applying (ii) repeatedly, we have
+
From (ii), note that
 
<cmath>\begin{alignat*}{8}
 
<cmath>\begin{alignat*}{8}
 
f(2) &= 1\cdot f(1) &&= 1, \\  
 
f(2) &= 1\cdot f(1) &&= 1, \\  
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and so on.
 
and so on.
  
−
In general, note that <cmath>f\left(2^n\right)=2^{(n-1)+(n-2)+\cdots+3+2+1}</cmath> for any positive integer <math>n.</math>
+
In general, we have <cmath>f\left(2^n\right)=2^{(n-1)+(n-2)+(n-3)+\cdots+3+2+1}</cmath> for any positive integer <math>n.</math>
−
 
 
−
Finally, the answer is <cmath>2^{100}=</cmath>
 
  
 +
Therefore, the answer is
 +
<cmath>\begin{align*}
 +
f\left(2^{100}\right)&=2^{99+98+97+\cdots+3+2+1} \\
 +
&=2^{99\cdot100/2} \\
 +
&= \boxed{\textbf {(D)}\ 2^{4950}}.
 +
\end{align*}</cmath>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
== Solution 2 (Backwards) ==
+
== Solution 2 (Backward) ==
−
We have
+
Applying (ii) repeatedly, we have
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
f\left(2^{100}\right) &= 2^{99} \cdot f\left(2^{99}\right) \\
 
f\left(2^{100}\right) &= 2^{99} \cdot f\left(2^{99}\right) \\
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&= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdot f\left(2^{97}\right) \\
 
&= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdot f\left(2^{97}\right) \\
 
&= \cdots \\
 
&= \cdots \\
−
&= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdots 2^{2} \cdot 2^{1} \cdot 1 \cdot f(1) \\
+
&= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdots 2^{3} \cdot 2^{2} \cdot 2^{1} \cdot 1 \cdot f(1) \\
−
&= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdots 2^{2} \cdot 2^{1} \cdot 1 \cdot 1 \\
+
&= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdots 2^{3} \cdot 2^{2} \cdot 2^{1} \cdot 1 \cdot 1 \\
−
&= 2^{99 + 98 + 97 + \cdots + 2 + 1} \\
+
&=2^{99 + 98 + 97 + \cdots + 3 + 2 + 1} \\
−
&= 2^{\frac{100(99)}{2}} \\
+
&=2^{99\cdot100/2} \\
−
&= \boxed{\text {(D)}\ 2^{4950}}
+
&= \boxed{\textbf {(D)}\ 2^{4950}}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
~Azjps (Fundamental Logic)
 
~Azjps (Fundamental Logic)

Latest revision as of 07:14, 6 September 2021

The following problem is from both the 2004 AMC 12A #17 and 2004 AMC 10A #24, so both problems redirect to this page.

Problem

Let $f$ be a function with the following properties:

(i) $f(1) = 1$, and

(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$.

What is the value of $f(2^{100})$?

$\textbf {(A)}\ 1 \qquad \textbf {(B)}\ 2^{99} \qquad \textbf {(C)}\ 2^{100} \qquad \textbf {(D)}\ 2^{4950} \qquad \textbf {(E)}\ 2^{9999}$

Solution 1 (Forward)

From (ii), note that \begin{alignat*}{8} f(2) &= 1\cdot f(1) &&= 1, \\ f\left(2^2\right) &= 2\cdot f(2) &&= 2, \\ f\left(2^3\right) &= 2^2\cdot f\left(2^2\right) &&= 2^{2+1}, \\ f\left(2^4\right) &= 2^3\cdot f\left(2^3\right) &&= 2^{3+2+1}, \end{alignat*} and so on.

In general, we have \[f\left(2^n\right)=2^{(n-1)+(n-2)+(n-3)+\cdots+3+2+1}\] for any positive integer $n.$

Therefore, the answer is \begin{align*} f\left(2^{100}\right)&=2^{99+98+97+\cdots+3+2+1} \\ &=2^{99\cdot100/2} \\ &= \boxed{\textbf {(D)}\ 2^{4950}}. \end{align*} ~MRENTHUSIASM

Solution 2 (Backward)

Applying (ii) repeatedly, we have \begin{align*} f\left(2^{100}\right) &= 2^{99} \cdot f\left(2^{99}\right) \\ &= 2^{99} \cdot 2^{98} \cdot f\left(2^{98}\right) \\ &= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdot f\left(2^{97}\right) \\ &= \cdots \\ &= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdots 2^{3} \cdot 2^{2} \cdot 2^{1} \cdot 1 \cdot f(1) \\ &= 2^{99} \cdot 2^{98} \cdot 2^{97} \cdots 2^{3} \cdot 2^{2} \cdot 2^{1} \cdot 1 \cdot 1 \\ &=2^{99 + 98 + 97 + \cdots + 3 + 2 + 1} \\ &=2^{99\cdot100/2} \\ &= \boxed{\textbf {(D)}\ 2^{4950}}. \end{align*} ~Azjps (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

Video Solution

https://youtu.be/qj5hBxYWalI

See also

2004 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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
2004 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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

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