1955 AMC 12 Problem 42

Question

If

a,

b,

and

c

are positive integers, the radicals

\sqrt{a+\dfrac bc}

and

a\sqrt{\dfrac bc}

are equal when and only when:

a=b=c=1

a=b

and

c=a=1

c=\dfrac{b(a^2-1)}a

a=b

and

c

is any value

a=b

and

c=a-1

Both sides are positive, so square the equality

Multiply the resulting equation by

c

and isolate

c

Because all quantities are positive, squaring is reversible:

a+\frac bc=a^2\frac bc.

Multiplying by

c

gives

ac+b=a^2b,

so

c=\frac{b(a^2-1)}a.

Thus, the correct answer is C.

Problem 42 in other years