2004 AMC 12A Problems/Problem 17

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(Redirected from 2004 AMC 10A Problems/Problem 24)

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:

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

$(ii)\quad f(2n) = n\times f(n)$ , for any positive integer $n$ .

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

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

Solution

$f(2^{100}) = f(2 \times 2^{99}) = 2^{99} \times f(2^{99})$ $= 2^{99} \cdot 2^{98} \times f(2^{98}) = \ldots$ $= 2^{99}2^{98}\cdots 2^{1} \cdot 1 \cdot f(1)$ $= 2^{99 + 98 + \ldots + 2 + 1}$ $= 2^{\frac{99(100)}{2}} = 2^{4950}$ $\Rightarrow \mathrm{(D)}$ .

2004 AMC 12A (Problems)
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

2004 AMC 10A (Problems)
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

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2004 AMC 12A Problems/Problem 17

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