1982 AMC 12 Problem 7

Attempt Problem 7 of the 1982 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 1982 AMC 12 solutions, or check the answer key.

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

If the operation

x*y

is defined by

x*y=(x+1)(y+1)-1,

then which one of the following is false?

x*y=y*x

for all real

x

and

y.

x*(y+z)

equals

(x*y)+(x*z)

for all real

x,

y,

and

z.

(x-1)*(x+1)

equals

(x*x)-1

for all real

x.

x*0=x

for all real

x.

x*(y*z)=(x*y)*z

for all real

x,

y,

and

z.

First simplify the operation to

x+y+xy

Test the proposed distributive law with

x=1

and

y=z=0

Since

x*y=x+y+xy,

the operation is commutative, has identity

0,

and becomes ordinary multiplication after adding

1.

Thus it is associative, and direct expansion also verifies C. Statement B fails when

x=1

and

y=z=0:

its left side is

1*0=1,

but its right side is

(1*0)+(1*0)=2.

Hence B is false.

Therefore, the correct answer is B.

Problem 7 in other years