2011 AMC 10A 第 19 题

Question

1991

年某镇人口是一个完全平方数。十年后，人口增加

150

人，变成比一个完全平方数多

9

。现在是

2011

年，人口又增加

150

人，再次成为完全平方数。下列哪一项最接近该镇这二十年人口增长的百分比？

In

1991

the population of a town was a perfect square. Ten years later, after an increase of

150

people, the population was

9

more than a perfect square. Now, in

2011,

with an increase of another

150

people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s population during this twenty-year period?

42

47

52

57

62

答案：E

知识点：平方差完全平方数百分数

难度评级：1750

小提示：

设

1991

年人口为

p^2

。

Let the

1991

population be

p^2

大提示：

使用

(q-p)(q+p)=141

，并检验因数对。

Use

(q-p)(q+p)=141

and test the factor pairs

解答：

设

1991

年人口为

p^2

。

2001

年人口为

q^2 + 9

。

于是

p^2 + 150 = q^2 + 9

q^2 - p^2 = 141\text{。}

因式分解得

(q - p)(q + p) = 141\text{。}

因为

p

和

q

都是整数，所以

q - p

与

q + p

的取值只可能是

(1, 141)

和

(3, 47)

。

检验第一对，得到

q-p=1

和

q+p=141\text{，}

两式相加后除以二，得到

q = 71

、

p = 70

。

但此时

p^2 + 300

不是完全平方数，所以这一对不合要求。

检验另一对，得到

q = 25

和

p = 22

。

此时

p^2 + 300 = 784

是完全平方数，人口增长率为

\dfrac{300}{22^2} \cdot 100 \% \approx 62 \%\text{。}

所以正确答案是 E。

Let the population in

1991

be

p^2.

Then let the population in

2001

be

q^2 + 9.

Using these values, we have

p^2 + 150 = q^2 + 9

q^2 - p^2 = 141.

Factoring, we get

(q - p)(q + p) = 141.

As

p

and

q

are integers, we have that the only possible values for

q - p

and

q + p

are

(1, 141)

and

(3, 47).

Trying the first pair, we have

q-p=1

and

q+p=141,

which adding together and dividing gives us

q = 71

and

p = 70.

We have that

p^2 + 300

is not a square number, which means that this pair is the wrong one.

Trying the other pair and using the same strategy gives us

q = 25

and

p = 22.

Now,

p^2 + 300 = 784,

which is a perfect square. The percent increase in population is then

\dfrac{300}{22^2} \cdot 100 \% \approx 62 \%.

Thus, E is the correct answer.

2011 AMC 10A 第 19 题

答案：E

难度评级：1750

小提示：

大提示：

解答：

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