2005 AMC 10A Problems/Problem 2: Difference between revisions

Latest revision as of 16:00, 1 July 2025

Problem

For each pair of real numbers $a \neq b$ , define the operation $\star$ as

$(a \star b) = \frac{a+b}{a-b}.$

What is the value of $((1 \star 2) \star 3)$ ?

$\textbf{(A) } -\frac{2}{3}\qquad \textbf{(B) } -\frac{1}{5}\qquad \textbf{(C) } 0\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \text{This value is not defined.}$

Solution

$((1 \star 2) \star 3) = \left(\left(\frac{1+2}{1-2}\right) \star 3\right) = (-3 \star 3) = \frac{-3+3}{-3-3} = \boxed{\textbf{(C) } 0}$ .

Video Solution 1

https://youtu.be/5g_m3_nck8E

Video Solution 2

https://youtu.be/6FnnFTWUJ0s

~Charles3829

2005 AMC 10A (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

@@ Line 1: / Line 1: @@
 ==Problem==
-For each pair of real numbers <math>a</math><math>\neq</math><math>b</math>, define the [[operation]] <math>\star</math> as
+For each pair of real numbers <math>a \neq b</math>, define the operation <math>\star</math> as
-<math> (a \star b) = \frac{a+b}{a-b} </math>.
+<cmath>(a \star b) = \frac{a+b}{a-b}.</cmath>
 What is the value of <math> ((1 \star 2) \star 3)</math>?
-<math> \mathrm{(A) \ } -\frac{2}{3}\qquad \mathrm{(B) \ } -\frac{1}{5}\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.} </math>
+<math>
+\textbf{(A) } -\frac{2}{3}\qquad \textbf{(B) } -\frac{1}{5}\qquad \textbf{(C) } 0\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \text{This value is not defined.}
+</math>
 ==Solution==
-<math> ((1 \star 2) \star 3) = \left(\left(\frac{1+2}{1-2}\right) \star 3\right) = (-3 \star 3) = \frac{-3+3}{-3-3} = 0 \Longrightarrow \mathrm{(C)}</math>
+<math> ((1 \star 2) \star 3) = \left(\left(\frac{1+2}{1-2}\right) \star 3\right) = (-3 \star 3) = \frac{-3+3}{-3-3} = \boxed{\textbf{(C) } 0}</math>.
-==See Also==
+==Video Solution 1==
-*[[2005 AMC 10A Problems]]
+https://youtu.be/5g_m3_nck8E
-*[[2005 AMC 10A Problems/Problem 1|Previous Problem]]
+==Video Solution 2==
+https://youtu.be/6FnnFTWUJ0s
-*[[2005 AMC 10A Problems/Problem 3|Next Problem]]
+~Charles3829
- [[Category:Introductory Algebra Problems]]
+==See also==
+{{AMC10 box|year=2005|ab=A|num-b=1|num-a=3}}
+[[Category:Introductory Number Theory Problems]]
+{{MAA Notice}}

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