2011 UNCO Math Contest II Problems/Problem 10
Contents
Problem
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.
Solution 1
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.
Solution 2
By using Simon's Favorite Factoring Trick, we can see that Something interesting happens when we set and \begin{align*} (m+1)(n+1)-1&=\\ (a-1+1)(b-1+1)-1&=\\ ab-1 \end{align*} If we represent each element of the set as the action repeatedly performed becomes much simpler. For a set performing the action stated in the problem until one value is left yields the value Setting to the first 50 integers and perform the action until one value is left, we get a value equal to Setting to the first integers yeilds a value equal to
See Also
2011 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |