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1993 AMC 12 #30 - Piecewise Function

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January 3, 2012
functioncalculusintegrationAMC

Problem Statement

Given 0≤x0<10 \le x_0 <1, let xn={2xn−1if 2xn−1<12xn−1−1if 2xn−1≥1 x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} for all integers n>0n>0. For how many x0x_0 is it true that x0=x5x_0=x_5?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 31<spanclass=′latex−bold′>(E)</span> infinitely many <span class='latex-bold'>(A)</span>\ 0 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ \text{infinitely many}
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