1966 AMC 12 Problem 22

Question

Consider the statements:

\text{(I) }\sqrt{a^2+b^2}=0

\text{(II) }\sqrt{a^2+b^2}=ab

\text{(III) }\sqrt{a^2+b^2}=a+b

\text{(IV) }\sqrt{a^2+b^2}=a-b

where we allow

a

and

b

to be real or complex numbers. Those statements for which there exist solutions other than

a=0

and

b=0,

are:

(\mathrm{I}),

(\mathrm{II}),

(\mathrm{III}),

(\mathrm{IV})

(\mathrm{II}),

(\mathrm{III}),

(\mathrm{IV})

only

(\mathrm{I}),

(\mathrm{III}),

(\mathrm{IV})

only

(\mathrm{III}),

(\mathrm{IV})

only

(\mathrm{I})

only

One nonzero example is enough for each statement

Try

b=i

for

(\mathrm{I}),

equal positive values for

(\mathrm{II}),

and

b=0

for

(\mathrm{III})

and

(\mathrm{IV})

Every statement has a nonzero solution. For

(\mathrm{I}),

use

(a,b)=(1,i).

For

(\mathrm{II}),

use

a=b=\sqrt2,

giving

\sqrt4=2=ab.

For both

(\mathrm{III})

and

(\mathrm{IV}),

use

(a,b)=(1,0).

Thus all four statements qualify.

Therefore, the correct answer is A.

Problem 22 in other years