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2004 AMC 12A Problems/Problem 17

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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^{999...

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)}.

See also

2004 AMC 12A (Problems • 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
2004 AMC 10A (Problems • 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
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