Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

Difference between revisions of "2004 AMC 10A Problems/Problem 2"
 
(6 intermediate revisions by 4 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
−
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:
+
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:
−
<math>
+
<cmath>\otimes(a,b,c)=\frac{a}{b-c}</cmath>
−
\otimes(a,b,c)=\frac{a}{b-c}
+
What is <math>\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))</math>?
−
</math>
 
−
What is <math>\otimes<cmath>( \otimes</cmath>(1,2,3),<cmath>\otimes</cmath>(2,3,1),<cmath>\otimes</cmath>(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>
 +
  
 
== Solution ==
 
== Solution ==
−
<math>\otimes<cmath> \left(\frac{1}{2-3}, \frac{2}{3-1}, \frac{3}{1-2}\right)=</cmath>\otimes</math><math>(-1,1,-3)=\frac{-1}{1+3}=-\frac{1}{4}\Longrightarrow\mathrm{(B)}</math>
+
<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>
 +
 
 +
 
 +
==Video Solution ==
 +
https://youtu.be/KfjB4--G-Lc
 +
 
 +
Education, the Study of Everything
 +
 
  
−
== See also ==
+
== 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]]
 
{{MAA Notice}}
 
{{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)
Preceded by
Problem 1 		Followed by
Problem 3
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 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_2&oldid=152240"