2011 AMC 10B Question 15

Let $@$ denote the “averaged with” operation: $a @ b = \frac{a+b}{2}.$ Which of the following distributive laws hold for all numbers $x,$ $y,$ and $z?$

I. $x @ (y + z) = (x @ y) + (x @ z)$

II. $x + (y @ z) = (x + y) @ (x + z)$

III. $x @ (y @ z) = (x @ y) @ (x @ z)$