2011 AMC 10B Problem 15

Question

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)

\mathrm{I}

only

\mathrm{II}

only

\mathrm{III}

only

\mathrm{I}

and

\mathrm{III}

only

\mathrm{II}

and

\mathrm{III}

only

Expand each side using

a@b=\frac{a+b}{2}

Test each of I, II, and III symbolically

In statement I, the left-hand side equals

\dfrac{x+y+z}2

and the right-hand side equals

x+\dfrac{y+z}2,

so they are not always equal.

In statement II, both sides equal

x+\dfrac{y+z}2,

so it holds.

In statement III, both sides equal

\dfrac{2x+y+z}4,

so it also holds.

Thus, the correct answer is E .

Problem 15 in other years