1953 AMC 12 第 45 题

Question

两条线段的长度分别为

a

个单位和

b

个单位。则它们之间的正确关系为：

The lengths of two line segments are

a

units and

b

units respectively. Then the correct relation between them is:

\dfrac{a+b}{2}\gt\sqrt{ab}

\dfrac{a+b}{2}\lt\sqrt{ab}

\dfrac{a+b}{2}=\sqrt{ab}

\dfrac{a+b}{2}\le\sqrt{ab}

\dfrac{a+b}{2}\ge\sqrt{ab}

从非负的平方

(a-b)^2

出发

Start with the nonnegative square

(a-b)^2

当两条线段等长时，等号必须可以成立

Equality must remain possible when the two segment lengths are equal

因为

(a-b)^2\ge0\text{，}

所以

(a+b)^2\ge4ab

。两边都非负，因此

\frac{a+b}{2}\ge\sqrt{ab}\text{。}

当

a=b

时等号成立。

因此，正确答案是 E。

Since

(a-b)^2\ge0,

we have

(a+b)^2\ge4ab.

Both sides are nonnegative, so

\frac{a+b}{2}\ge\sqrt{ab}.

Equality occurs when

a=b.

Thus, the correct answer is E.

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