Difference between revisions of "2011 UNCO Math Contest II Problems/Problem 10"
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== Solution == | == Solution == | ||
| − | + | First try <math>\{1, 2, 3, \ldots , n\}</math> for <math>n= 2, 3, 4, 5</math>. The crossing off process yields <math>\{5,23,119,719\}</math> each one being one less than a factorial. So for general <math>n</math> you should end up with<math>(n+1)!-1</math>. Now look at <math>n=3</math> again and replace <math>1, 2, 3</math> with <math>a,b,c</math> (order does not matter). Crossing off gives you <cmath>(a+b+ab) + c + (a+b+ab)c =a+b+c+ab+ac+bc+abc</cmath> reminding one of the coefficients in <cmath>(x-a)(x-b)(x-c)= x^3-(a+b+c)x^2+(ab+ac+bc)x-abc</cmath> Now let <math>x=-1</math>, and watch what happens remember that <math>\{a,b,c\} = \{1,2,3\}</math>. There are other approaches. | |
== See Also == | == See Also == | ||
Revision as of 02:23, 13 January 2019
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
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 ![\[(a+b+ab) + c + (a+b+ab)c =a+b+c+ab+ac+bc+abc\]](http://latex.artofproblemsolving.com/d/f/3/df32bf0089ce03bf07b1428033a04eff3c24eb46.png)
reminding one of the coefficients in ![\[(x-a)(x-b)(x-c)= x^3-(a+b+c)x^2+(ab+ac+bc)x-abc\]](http://latex.artofproblemsolving.com/9/7/c/97c99449bfa0c5bc6f98d38dc412d010d8d84de0.png)
Now let 
, and watch what happens remember that 
. There are other approaches.
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 | ||
