Problem #2134

 2134 Let $a_1,a_2,\cdots$, be a sequence with the following properties. (i) $a_1=1$, and (ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$. What is the value of $a_{2^{100}}$? $\mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2^{99} \qquad \mathrm{(C) \ } 2^{100} \qquad \mathrm{(D) \ } 2^{4950} \qquad \mathrm{(E) \ } 2^{9999}$ This problem is copyrighted by the American Mathematics Competitions.
Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved.

• Reduce fractions to lowest terms and enter in the form 7/9.
• Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
• Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
• Exponents should be entered in the form 10^10.
• If the problem is multiple choice, enter the appropriate (capital) letter.
• Enter points with parentheses, like so: (4,5)
• Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

Registration open for 2018 AIME prep class

Registration is now open. See details here.

\$25 discount if you register by December 28!