1955 AMC 12 第 42 题

Question

若

a

、

b

和

c

均为正整数，则根式

\sqrt{a+\dfrac bc}

与

a\sqrt{\dfrac bc}

相等的充要条件是：

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

且

c=a=1

a=b

and

c=a=1

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

a=b

且

c

可取任意值

a=b

and

c

is any value

a=b

且

c=a-1

a=b

and

c=a-1

等式两边均为正数，所以将两边平方

Both sides are positive, so square the equality

将所得方程乘以

c

，再解出

c

Multiply the resulting equation by

c

and isolate

c

由于所有量均为正数，平方是等价变形：

a+\frac bc=a^2\frac bc\text{。}

两边乘以

c

，得

ac+b=a^2b

，所以

c=\frac{b(a^2-1)}a\text{。}

因此，正确答案是 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.

其他年份的第 42 题