1962 AMC 12 第 27 题

Question

令

a\mathbin{@}b

表示对两个数

a

和

b

取较大者的运算，并规定

a\mathbin{@}a=a

。令

a\mathbin{!}b

表示取两数中较小者的运算，并规定

a\mathbin{!}a=a

。以下三条法则中哪些正确？

\begin{aligned} (1)\quad&a\mathbin{@}b=b\mathbin{@}a,\\ (2)\quad&a\mathbin{@}(b\mathbin{@}c)=(a\mathbin{@}b)\mathbin{@}c,\\ (3)\quad&a\mathbin{!}(b\mathbin{@}c)\\ &\quad=(a\mathbin{!}b)\mathbin{@}(a\mathbin{!}c) \end{aligned}\text{。}

Let

a\mathbin{@}b

represent the operation on two numbers,

a

and

b,

which selects the larger of the two numbers, with

a\mathbin{@}a=a.

Let

a\mathbin{!}b

represent the operation which selects the smaller of the two numbers, with

a\mathbin{!}a=a.

Which of the following three rules is (are) correct?

\begin{aligned} (1)\quad&a\mathbin{@}b=b\mathbin{@}a,\\ (2)\quad&a\mathbin{@}(b\mathbin{@}c)=(a\mathbin{@}b)\mathbin{@}c,\\ (3)\quad&a\mathbin{!}(b\mathbin{@}c)\\ &\quad=(a\mathbin{!}b)\mathbin{@}(a\mathbin{!}c). \end{aligned}

仅

(1)

(1)

only

仅

(2)

(2)

only

仅

(1)

和

(2)

(1)

and

(2)

only

仅

(1)

和

(3)

(1)

and

(3)

only

三条都正确

all three

答案：E

知识点：自定义运算分配律分类讨论

难度评级：1500

小提示：

把

\mathbin{@}

理解为取最大值，把

\mathbin{!}

理解为取最小值

Translate

\mathbin{@}

as maximum and

\mathbin{!}

as minimum

大提示：

对法则

(3)

，分别在

a

小于或大于

\max(b,c)

时比较两边

For rule

(3),

compare both sides separately when

a

is below or above

\max(b,c)

解答：

取最大值满足交换律和结合律，所以

(1)

与

(2)

成立。法则

(3)

是分配恒等式

\begin{aligned} &\min(a,\max(b,c))\\ &\quad=\max(\min(a,b),\min(a,c)) \end{aligned}\text{。}

若

a\ge\max(b,c)

，两边都等于

\max(b,c)

；若

a\lt\max(b,c)

，两边都等于

a

。因此

(3)

也成立。

所以正确答案是 E。

Maximum is commutative and associative, so

(1)

and

(2)

hold. Rule

(3)

is the distributive identity

\begin{aligned} &\min(a,\max(b,c))\\ &\quad=\max(\min(a,b),\min(a,c)). \end{aligned}

If

a\ge\max(b,c),

both sides equal

\max(b,c);

if

a\lt\max(b,c),

both sides equal

a.

Thus

(3)

also holds.

Therefore, the correct answer is E.

1962 AMC 12 第 27 题

答案：E

难度评级：1500

小提示：

大提示：

解答：

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