Difference between revisions of "2000 AMC 12 Problems/Problem 12"
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== Solution == | == Solution == | ||
| − | + | It is not hard to see that | |
| + | <cmath>(A+1)(M+1)(C+1)=AMC+AM+AC+MC+A+M+C+1</cmath> | ||
| + | Since <math>A+M+C=12</math>, we can rewrite this as | ||
| + | <cmath>AMC+AM+AC+MC+13</cmath> | ||
| + | So we wish to maximize | ||
| + | <cmath>(A+1)(M+1)(C+1)-13</cmath> | ||
| + | Which is largest when all the factors are equal. Since <math>A+M+C=12</math>, we set <math>A=B=C=4</math> | ||
| + | Which gives us | ||
| + | <cmath>(4+1)(4+1)(4+1)-13=112</cmath> | ||
| + | so the answer is <math>\boxed{E}</math>. | ||
== See also == | == See also == | ||
Revision as of 14:39, 16 May 2015
Problem
Let 
and 
be nonnegative integers such that 
. What is the maximum value of 
?
Solution
It is not hard to see that
![\[(A+1)(M+1)(C+1)=AMC+AM+AC+MC+A+M+C+1\]](http://latex.artofproblemsolving.com/9/d/1/9d1e2c3f70df8b68b48df6e75068757237c4ef18.png)
Since 
, we can rewrite this as
![\[AMC+AM+AC+MC+13\]](http://latex.artofproblemsolving.com/7/9/f/79f3f8b8cb3dee37405c83c414548b1ebd2a63c5.png)
So we wish to maximize
![\[(A+1)(M+1)(C+1)-13\]](http://latex.artofproblemsolving.com/a/3/f/a3f9f9ef8289341a9a4f3e43f0aa4199a2d28b58.png)
Which is largest when all the factors are equal. Since , we set 
Which gives us
![\[(4+1)(4+1)(4+1)-13=112\]](http://latex.artofproblemsolving.com/8/c/c/8cc110b16d54e7fd301a03ee24ee13a2f40e0e05.png)
so the answer is 
.
See also
| 2000 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
