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| | == Solution == | | == Solution == |
| − | ===Solution 1===
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| − | There are 8 [[fraction]]s which fit the conditions between 0 and 1: <math>\frac{1}{30},\frac{7}{30},\frac{11}{30},\frac{13}{30},\frac{17}{30},\frac{19}{30},\frac{23}{30},\frac{29}{30}</math>
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| − | Their sum is 4. Note that there are also 8 terms between 1 and 2 which we can obtain by adding 1 to each of our first 8 terms. For example, <math>1+\frac{19}{30}=\frac{49}{30}.</math> Following this pattern, our answer is <math>4(10)+8(1+2+3+\cdots+9)=\boxed{400}.</math>
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| − | ===Solution 2===
| + | This is cool. Thanks Bye |
| − | By [[Euler's Totient Function]], there are <math>8</math> numbers that are relatively prime to <math>30</math>, less than <math>30</math>. Note that they come in pairs <math>(m,30-m)</math> which result in sums of <math>1</math>; thus the sum of the smallest <math>8</math> rational numbers satisfying this is <math>\frac12\cdot8\cdot1=4</math>. Now refer to solution 1.
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| − | === Solution 3===
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| − | Note that if <math>x</math> is a solution, then <math>(300-x)</math> is a solution. We know that <math>\phi(300) = 80.</math> Therefore the answer is <math>\displaystyle\frac{80}{2} \cdot \displaystyle\frac{300}{30} = \boxed{400}.</math>
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| − | {{AIME box|year=1992|before=First question|num-a=2}}
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| − | {{MAA Notice}}
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Revision as of 19:59, 23 February 2022
