Prove that there is a unique maximal consistent cut preceding any given cut. Also, prove that algorithm for finding maximal consistent cut is correct.

1)In the tournament mutual exclusion algorithm, given n threads, how many pair-wise competitions are needed to be performed until every thread enters the critical section? Assume that each thread enters the critical section exactly once

2) Prove that there is a unique maximal consistent cut preceding any given cut. Also, prove that algorithm for finding maximal consistent cut is correct.

3)Consider a synchronous system. Suppose that we want Byzantine consensus to satisfy the following condition: every non-faulty node’s decision must equal the input of some non-faulty processor. Show that to be able to satisfy these conditions while tolerating up to f-Byzantine faults, the number of nodes n must be at least max(3, m)f+ 1, if each node’s input is an integer value between 0 and m-1 and m > 2

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