MathDB
"Averaged with" operation

Source: AMC 10 2011 b Problem 15

February 24, 2011
AMC

Problem Statement

Let @ denote the "averaged with" operation: aa @ bb = a+b2\frac{a+b}{2}. Which of the following distributive laws hold for all numbers x,yx,y and zz?
I. x @ (y+z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z)
<spanclass=′latex−bold′>(A)</span> I only<spanclass=′latex−bold′>(B)</span> II only<spanclass=′latex−bold′>(C)</span> III only<spanclass=′latex−bold′>(D)</span> I and III only<spanclass=′latex−bold′>(E)</span> II and III only <span class='latex-bold'>(A)</span>\ \text{I only} \qquad <span class='latex-bold'>(B)</span>\ \text{II only} \qquad <span class='latex-bold'>(C)</span>\ \text{III only} \qquad <span class='latex-bold'>(D)</span>\ \text{I and III only} \qquad <span class='latex-bold'>(E)</span>\ \text{II and III only}
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