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Difference between revisions of "2004 AMC 10A Problems/Problem 2"

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== Problem ==
 
== Problem ==
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For any three [[real number]]s <math>a</math>, <math>b</math>, and <math>c</math>, with <math>b\neq c</math>, the [[operation]] <math>\otimes</math> is defined by:
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For any three real numbers <math>a</math>, <math>b</math>, and <math>c</math>, with <math>b\neq c</math>, the operation <math>\otimes</math> is defined by:
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<math>
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<cmath>\otimes(a,b,c)=\frac{a}{b-c}</cmath>
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\otimes(a,b,c)=\frac{a}{b-c}
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What is <math>\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))</math>?
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</math>
 
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What is <math>\otimes</math><math>( \otimes</math><math>(1,2,3),</math><math>\otimes</math><math>(2,3,1),</math><math>\otimes</math><math>(3,1,2))</math>?
 
  
 
<math> \mathrm{(A) \ } -\frac{1}{2}\qquad \mathrm{(B) \ } -\frac{1}{4} \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } \frac{1}{4} \qquad \mathrm{(E) \ } \frac{1}{2} </math>
 
<math> \mathrm{(A) \ } -\frac{1}{2}\qquad \mathrm{(B) \ } -\frac{1}{4} \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } \frac{1}{4} \qquad \mathrm{(E) \ } \frac{1}{2} </math>
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== Solution ==
 
== Solution ==
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<math>\otimes</math><math> \left(\frac{1}{2-3}, \frac{2}{3-1}, \frac{3}{1-2}\right)=\displaystyle</math><math>\otimes</math><math>(-1,1,-3)=\frac{-1}{1+3}=-\frac{1}{4}\Longrightarrow\mathrm{(B)}</math>
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<math>\otimes \left(\frac{1}{2-3},\frac{2}{3-1},\frac{3}{1-2}\right)=\otimes(-1,1,-3)=\frac{-1}{1+3}=-\frac{1}{4}\Longrightarrow\boxed{\mathrm{(B)}\ -\frac{1}{4}}</math>
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==Video Solution ==
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https://youtu.be/KfjB4--G-Lc
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Education, the Study of Everything
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== See also ==
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== See Also ==
 
{{AMC10 box|year=2004|ab=A|num-b=1|num-a=3}}
 
{{AMC10 box|year=2004|ab=A|num-b=1|num-a=3}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 13:13, 21 April 2021

Problem

For any three real numbers $a$, $b$, and $c$, with $b\neq c$, the operation $\otimes$ is defined by: \[\otimes(a,b,c)=\frac{a}{b-c}\] What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$?

$\mathrm{(A) \ } -\frac{1}{2}\qquad \mathrm{(B) \ } -\frac{1}{4} \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } \frac{1}{4} \qquad \mathrm{(E) \ } \frac{1}{2}$


Solution

$\otimes \left(\frac{1}{2-3},\frac{2}{3-1},\frac{3}{1-2}\right)=\otimes(-1,1,-3)=\frac{-1}{1+3}=-\frac{1}{4}\Longrightarrow\boxed{\mathrm{(B)}\ -\frac{1}{4}}$


Video Solution

https://youtu.be/KfjB4--G-Lc

Education, the Study of Everything


See Also

2004 AMC 10A (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
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All AMC 10 Problems and Solutions

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