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3

Part of 1976 Putnam

Problems(2)

Putnam 1976 A3

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4/18/2022
Find all integral solutions of the equation ∣pr−qs∣=1,|p^r-q^s|=1, where pp and qq are prime numbers and rr and ss are positive integers larger than unity. Prove that there are no other solutions.
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Putnam 1976 B3

Source:

4/20/2022
Suppose that we have nn events A1,…,An,A_1,\dots, A_n, each of which has probability at least 1−a1-a of occuring, where a<1/4.a<1/4. Further suppose that AiA_i and AjA_j are mutually independent if ∣i−j∣>1.|i-j|>1. Assume as known that the recurrence uk+1=uk−auk−1,u0=1,u1=1−a,u_{k+1}=u_k-au_{k-1}, u_0=1, u_1=1-a, defines positive real numb uku_k for k=0,1,… .k=0,1,\dots. Show that the probability of all of A1,…,AnA_1,\dots, A_n occuring is at least un.u_n.
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