Difference between revisions of "2011 AMC 10B Problems/Problem 15"
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Let <math>@</math> denote the "averaged with" operation: <math>a @ b = \frac{a+b}{2}</math>. Which of the following distributive laws hold for all numbers <math>x, y,</math> and <math>z</math>? <cmath>\text{I. x @ (y + z) = (x @ y) + (x @ z)}</cmath> <cmath>\text{II. x + (y @ z) = (x + y) @ (x + z)}</cmath> <cmath>\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}</cmath> | Let <math>@</math> denote the "averaged with" operation: <math>a @ b = \frac{a+b}{2}</math>. Which of the following distributive laws hold for all numbers <math>x, y,</math> and <math>z</math>? <cmath>\text{I. x @ (y + z) = (x @ y) + (x @ z)}</cmath> <cmath>\text{II. x + (y @ z) = (x + y) @ (x + z)}</cmath> <cmath>\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}</cmath> | ||
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<math>\boxed{\textbf{(E)} \text{II and III only}}</math> | <math>\boxed{\textbf{(E)} \text{II and III only}}</math> | ||
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+ | == See Also== | ||
+ | |||
+ | {{AMC10 box|year=2011|ab=B|num-b=14|num-a=16}} |
Revision as of 17:14, 4 June 2011
Problem
Let denote the "averaged with" operation:
. Which of the following distributive laws hold for all numbers
and
?
Solution
Just simplify each operation and see which ones hold true.
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 |