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| − | == Problem ==
| + | #redirect [[2010 AMC 12B Problems/Problem 19]] |
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| − | == Solution ==
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| − | Represent the teams' scores as: <math>(a, an, an^2, an^3)</math> and <math>(a, a+m, a+2m, a+3m)</math>
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| − | We have <math>a+an+an^2+an^3=4a+6m+1</math>
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| − | Manipulating this, we can get <math>a(1+n+n^2+n^3)=4a+6m+1</math>, or <math>a(n^4-1)/(n-1)=4a+6m+1</math>
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| − | Since both are increasing sequences, <math>n>1</math>. We can check cases up to <math>n=4</math> because when <math>n=5</math>, we get <math>156a>100</math>. When
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| − | * <math>n=2, a=[1,6]</math>
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| − | *<math> n=3, a=[1,2]</math>
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| − | *<math> n=4, a=1</math>
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| − | Checking each of these cases individually back into the equation <math>a+an+an^2+an^3=4a+6m+1</math>, we see that only when a=5 and n=2, we get an integer value for m, which is 9. The original question asks for the first half scores summed, so we must find <math>(a)+(an)+(a)+(a+m)=(5)+(10)+(5)+(5+9)=34</math>
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| − | == See also ==
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| − | {{AMC10 box|year=2010|ab=B|num-b=23|num-a=25}}
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