1973 AMC 12 Problem 5

Question

Of the following five statements,

\mathrm{I}

to

\mathrm{V},

about the binary operation of averaging (arithmetic mean),

\mathrm{I}.

Averaging is associative

\mathrm{II}.

Averaging is commutative

\mathrm{III}.

Averaging distributes over addition

\mathrm{IV}.

Addition distributes over averaging

\mathrm{V}.

Averaging has an identity element

those which are always true are

All

\mathrm{I}

and

\mathrm{II}

only

\mathrm{II}

and

\mathrm{III}

only

\mathrm{II}

and

\mathrm{IV}

only

\mathrm{II}

and

\mathrm{V}

only

Write the averaging operation as

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

Expand each proposed identity; one counterexample is enough to reject a property

Let

a*b=\frac{a+b}{2}.

Commutativity is immediate. Also

\begin{aligned} a+(b*c) &=\frac{2a+b+c}{2}\\ &=(a+b)*(a+c). \end{aligned}

so addition distributes over averaging.

Associativity fails because

\begin{aligned} (a*b)*c&=\frac{a+b+2c}{4},\\ a*(b*c)&=\frac{2a+b+c}{4}. \end{aligned}

Averaging does not distribute over addition since generally

a*(b+c)\ne(a*b)+(a*c).

Finally, an identity

e

would require

\frac{e+a}{2}=a

for every

a,

or

e=a,

which is impossible for a fixed

e.

Thus only

\mathrm{II}

and

\mathrm{IV}

always hold.

Therefore, the correct answer is D.

Problem 5 in other years