2011 UNCO Math Contest II Problems/Problem 10
The integers are written on the blackboard. Select any two, call them and and replace these two with the one number . Continue doing this until only one number remains and explain, with proof, what happens. Also explain with proof what happens in general as you replace with . As an example, if you select and you replace them with . If you select and , replace them with . You now have two ’s in this case but that’s OK.
First try for . The crossing off process yields each one being one less than a factorial. So for general you should end up with. Now look at again and replace with (order does not matter). Crossing off gives you reminding one of the coefficients in Now let , and watch what happens remember that . There are other approaches.
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