2013 AMC 12B 第 21 题

Question

考虑如下定义的

30

条抛物线：所有抛物线的焦点都是

(0, 0)

，准线均形如

y = ax + b

，其中

a

和

b

是整数，且

a \in \{-2, -1, 0, 1, 2\}

、

b \in \{-3, -2, -1, 1, 2, 3\}

。这些抛物线中没有三条有公共点。平面上有多少个点恰好位于其中两条抛物线上？

Consider the set of

30

parabolas defined as follows: all parabolas have as focus the point

(0, 0)

and the directrix lines have the form

y = ax + b

with

a

and

b

integers such that

a \in \{-2, -1, 0, 1, 2\}

and

b \in \{-3, -2, -1, 1, 2, 3\}.

No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?

720

760

810

840

870

答案：C

知识点：抛物线补集计数组合

难度评级：2360

小提示：

两条有共同焦点的抛物线通常交于

2

点；例外是准线平行且焦点不在两准线之间。

Two parabolas sharing a focus meet in

2

points unless their directrices are parallel with the focus not between them

大提示：

从所有

\binom{30}{2}

对中减去不相交的对：斜率相同且

y

-截距同号。

From all

\binom{30}{2}

pairs, subtract the non-intersecting ones: equal slope and same-sign

y

-intercepts

解答：

两条共同焦点为

O

的抛物线恰有

2

个交点，除非它们的准线平行且

O

在两准线之间的带状区域外，此时不相交。不相交的对具有相同斜率，且

y

-截距同号。共有

5

个斜率；对每个斜率，同号截距对有

2\binom{3}{2} = 6

对。每个相交对贡献

2

个点，且没有三条共点，所以总数为

2\left(\binom{30}{2} - 5\cdot 6\right)

= 2(435 - 30)

= 810

。所以正确答案是 C。

Two parabolas with common focus

O

meet in exactly

2

points, except when their directrices are parallel and

O

lies outside the strip between them, in which case they do not meet. The non-intersecting pairs have directrices of equal slope and

y

-intercepts of the same sign. There are

5

slopes, and for each,

2\binom{3}{2} = 6

same-sign intercept pairs. Since every intersecting pair meets in

2

points and no point lies on three parabolas, the total is

2\left(\binom{30}{2} - 5\cdot 6\right)

= 2(435 - 30)

= 810.

Thus, the correct answer is C.

2013 AMC 12B 第 21 题

答案：C

难度评级：2360

小提示：

大提示：

解答：

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