1986 IMO Problems/Problem 3
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers respectively, and , then the following operation is allowed: are replaced by respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
The algorithm always stops. Consider the functionClearly always and is integer valued. Suppose, WLOG, that . Then since . Thus if the algorithm does not stop, we can find an infinite decreasing sequence of nonnegative integers . This is impossible, so the algorithm must stop.
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