Difference between revisions of "2011 AMC 10B Problems/Problem 15"
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== Solution == | == Solution == | ||
− | + | Simplify each operation and see which ones hold true. | |
− | |||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
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\text{III.} \qquad x @ (y @ z) &= (x @ y) @ (x @ z)\\ | \text{III.} \qquad x @ (y @ z) &= (x @ y) @ (x @ z)\\ | ||
x @ \frac{y+z}{2} &= \frac{x+y}{2} @ \frac{x+z}{2}\\ | x @ \frac{y+z}{2} &= \frac{x+y}{2} @ \frac{x+z}{2}\\ | ||
− | \frac{2x+y+z}{ | + | \frac{2x+y+z}{4} &= \frac{2x+y+z}{4} |
\end{align*}</cmath> | \end{align*}</cmath> | ||
<math>\boxed{\textbf{(E)} \text{II and III only}}</math> | <math>\boxed{\textbf{(E)} \text{II and III only}}</math> | ||
+ | |||
+ | ==Solution 2== | ||
+ | Alternatively, substitute arbitrary values for <math>x</math>, <math>y</math>, and <math>z</math> where <math>x\neq y\neq z</math>. For example, <math>x=1</math>, <math>y=2</math>, and <math>z=3</math> give that only II and III work. <math>\boxed{\textbf{(E)} \text{II and III only}}</math> | ||
== See Also== | == See Also== | ||
{{AMC10 box|year=2011|ab=B|num-b=14|num-a=16}} | {{AMC10 box|year=2011|ab=B|num-b=14|num-a=16}} | ||
+ | {{MAA Notice}} |
Latest revision as of 01:48, 26 June 2022
Contents
Problem
Let denote the "averaged with" operation:
. Which of the following distributive laws hold for all numbers
and
?
Solution
Simplify each operation and see which ones hold true.
Solution 2
Alternatively, substitute arbitrary values for ,
, and
where
. For example,
,
, and
give that only II and III work.
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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 |
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