1971 AMC 12 Problem 6

Attempt Problem 6 of the 1971 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1971 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Let

*

be a symbol denoting the binary operation on the set

S

of all nonzero real numbers as follows: for any

a,

b\in S,

a*b=2ab.

Which statement is not true?

*

is commutative over

S

*

is associative over

S

\frac{1}{2}

is an identity element for

*

S

Every element of

S

has an inverse for

*

\frac{1}{2a}

is an inverse for

*

of the element

a

S

First determine the identity

e

from

a*e=a

An inverse

x

a

must satisfy

a*x=\frac{1}{2}

The operation is commutative, and

(a*b)*c=4abc=a*(b*c),

so it is associative. Its identity is

\frac{1}{2}.

The inverse of

a

must satisfy

2ax=\frac{1}{2},

so it is

x=\frac{1}{4a},

not

\frac{1}{2a}.

Therefore, the correct answer is E.

Problem 6 in other years