2013 AMC 10B 第 5 题

Question

正整数

a

和

b

都小于

6

。求下式的最小可能值：

2 \cdot a - a \cdot b

Positive integers

a

and

b

are each less than

6.

What is the smallest possible value of the following expression?

2 \cdot a - a \cdot b

-20

-15

-10

0

2

答案：B

知识点：因式分解最优化

难度评级：870

小提示：

把表达式因式分解为

a(2-b)

。

Factor the expression as

a(2-b)

大提示：

让

2-b

尽可能负，同时让

a

尽可能大。

Make

2-b

as negative as possible while making

a

large

解答：

2a-ab=a(2-b)

。

为了使它最小，取

b

尽可能大，使

2-b

最负，同时取

a

尽可能大。因为

a,b<6

，可取

a=b=5

。

此时

5(2-5)=-15

。

所以正确答案是 B。

The expression is

2a-ab=a(2-b)

.

To make it as small as possible, choose

b

as large as possible so that

2-b

is most negative, and choose

a

as large as possible. Since

a,b<6

, take

a=b=5

.

The minimum value is

5(2-5)=-15

.

Thus, the correct answer is B .

2013 AMC 10B 第 5 题

答案：B

难度评级：870

小提示：

大提示：

解答：

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