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Sample Question 24:
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 \(f(2^{100})\)?
\(\mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2^{99} \qquad \mathrm{(C) \ } 2^{100} \qquad \mathrm{(D) \ } 2^{4950} \qquad \mathrm{(E) \ } 2^{9999}\)