MathDB
2022 Putnam B5

Source:

December 4, 2022
PutnamPutnam 2022

Problem Statement

For 0≤p≤1/2,0 \leq p \leq 1/2, let X1,X2,…X_1, X_2, \ldots be independent random variables such that Xi={1with probability p,−1with probability p,0with probability 1−2p,X_i=\begin{cases} 1 & \text{with probability } p, \\ -1 & \text{with probability } p, \\ 0 & \text{with probability } 1-2p, \end{cases} for all i≥1.i \geq 1. Given a positive integer nn and integers b,a1,…,an,b,a_1, \ldots, a_n, let P(b,a1,…,an)P(b, a_1, \ldots, a_n) denote the probability that a1X1+…+anXn=b.a_1X_1+ \ldots + a_nX_n=b. For which values of pp is it the case that P(0,a1,…,an)≥P(b,a1,…,an)P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n) for all positive integers nn and all integers b,a1,…,an?b, a_1,\ldots, a_n?
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