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| == Problem ==
| | НИГЕР |
| <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Let <math>A, M,</math> and <math>C</math> be [[nonnegative integer]]s such that <math>A + M + C=12</math>. What is the maximum value of <math>A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude>
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| <math> \mathrm{(A) \ 62 } \qquad \mathrm{(B) \ 72 } \qquad \mathrm{(C) \ 92 } \qquad \mathrm{(D) \ 102 } \qquad \mathrm{(E) \ 112 } </math>
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| == Solution 1 ==
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| It is not hard to see that
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| <cmath>(A+1)(M+1)(C+1)=</cmath>
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| <cmath>AMC+AM+AC+MC+A+M+C+1</cmath>
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| Since <math>A+M+C=12</math>, we can rewrite this as
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| <cmath>(A+1)(M+1)(C+1)=</cmath>
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| <cmath>AMC+AM+AC+MC+13</cmath>
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| So we wish to maximize
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| <cmath>(A+1)(M+1)(C+1)-13</cmath>
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| Which is largest when all the factors are equal (consequence of AM-GM). Since <math>A+M+C=12</math>, we set <math>A=M=C=4</math>
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| Which gives us
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| <cmath>(4+1)(4+1)(4+1)-13=112</cmath>
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| so the answer is <math>\boxed{\text{E}}</math>.
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| I wish you understand this problem and can use it in other problems.
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| == Solution 2 (Nonrigorous) ==
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| If you know that to maximize your result you <math>\textit{usually}</math> have to make the numbers as close together as possible, (for example to maximize area for a polygon make it a square) then you can try to make <math>A,M</math> and <math>C</math> as close as possible. In this case, they would all be equal to <math>4</math>, so <math>AMC+AM+AC+MC=64+16+16+16=112</math>, giving you the answer of <math>\boxed{\text{E}}</math>.
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| == Solution 3 ==
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| Assume <math>A</math>, <math>M</math>, and <math>C</math> are equal to <math>4</math>. Since the resulting value of <math>AMC+AM+AC+MC</math> will be <math>112</math> and this is the largest answer choice, our answer is <math>\boxed{\textbf{(E) }112}</math>.
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| == Video Solution ==
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| https://youtu.be/lxqxQhGterg
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| == See also ==
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| {{AMC12 box|year=2000|num-b=11|num-a=13}}
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| [[Category:Introductory Algebra Problems]]
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| {{MAA Notice}}
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