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2011 AMC 10B Problem 15

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2011 AMC 10B Problem 15AMC 10 Step by Step
Name:                             Date: September 13, 2026
Functions · difficulty 2 of 5 · about 3 min · position P11-15

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

  1. A)

    I only\text{I only}

  2. B)

    II only\text{II only}

  3. C)

    III only\text{III only}

  4. D)

    I and III only\text{I and III only}

  5. E)

    II and III only\text{II and III only}

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