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A4

Part of 2022 Putnam

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2022 Putnam A4

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12/4/2022
Suppose that X1,X2,…X_1, X_2, \ldots are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let S=∑i=1kXi/2i,S=\sum_{i=1}^kX_i/2^i, where kk is the least positive integer such that Xk<Xk+1,X_k<X_{k+1}, or k=∞k=\infty if there is no such integer. Find the expected value of S.S.
PutnamPutnam 2022