Problem #1667


Let $@$ denote the "averaged with" operation: $a @ b = (a+b)/2$. Which of the following distributive laws hold for all numbers $x, y,$ and $z$? \[\text{I. x @ (y + z) = (x @ y) + (x @ z)}\] \[\text{II. x + (y @ z) = (x + y) @ (x + z)}\] \[\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}\]

$\textbf{(A)}\ \text{I only} \qquad\textbf{(B)}\ \text{II only} \qquad\textbf{(C)}\ \text{III only} \qquad\textbf{(D)}\ \text{I and III only} \qquad\textbf{(E)}\ \text{II and III only}$

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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