2014 AMC 12B 第 24 题

先试着解答 2014 AMC 12B 第 24 题，然后核对你的答案与精心整理的解答，解答来自 LIVE by Po-Shen Loh。你也可以参加完整限时模拟考试、查看全部 2014 AMC 12B 解答，或核对答案。

所有题目均经美国数学协会（MAA）官方合法授权使用。

24.

设

ABCDE

是一个内接于圆的五边形，满足

AB = CD = 3

，

BC = DE = 10

，且

AE = 14

。

ABCDE

所有对角线长度之和等于

\dfrac{m}{n}

，其中

m

和

n

是互质正整数。

m + n

是多少？

Let

ABCDE

be a pentagon inscribed in a circle such that

AB = CD = 3,

BC = DE = 10,

and

AE = 14.

The sum of the lengths of all diagonals of

ABCDE

is equal to

\dfrac{m}{n},

where

m

and

n

are relatively prime positive integers. What is

m + n?

129

247

353

391

421

答案：D

知识点：托勒密定理弦多项式

难度评级：2650

小提示：

等弦所对的弧相等，所以

AC = BD = CE

；设这个公共长度为

x

。

Equal chords subtend equal arcs, so

AC = BD = CE;

call this

x

大提示：

对

ABCD

，

BCDE

，和

ABDE

使用托勒密定理，得到关于

x, y = AD, z = BE

的方程。

Apply Ptolemy’s theorem to

ABCD,

BCDE,

and

ABDE

to get equations in

x, y = AD, z = BE

解答：

因为弧

AB, CD

相等，弧

BC, DE

相等，所以弦

AC, BD, CE

都相等；设

x = AC = BD = CE

，

y = AD

，且

z = BE

。

对

ABCD

、

BCDE

和

ABDE

使用托勒密定理，得到

\begin{gathered} 10y + 9 = x^2, \\ \quad 100 + 3z = x^2, \\ \quad 30 + 14x = yz\text{。} \end{gathered}

由前两个方程解出

y

和

z

，再代入第三个方程，得到

\begin{gathered} x^3 - 109x - 420 = 0 \\ = (x-12)(x+5)(x+7)\text{。} \end{gathered}

所以

x = 12

，

y = \tfrac{135}{10} = \tfrac{27}{2}

，且

z = \tfrac{44}{3}

。五条对角线为

AC, BD, CE, AD, BE

，它们的和为

\begin{gathered} 3x + y + z = 36 \\ {}+ \tfrac{27}{2} + \tfrac{44}{3} \\ = \tfrac{385}{6}\text{。} \end{gathered}

因此

m + n = 385 + 6 = 391

，正确答案是 D。

Because arcs

AB, CD

are equal and arcs

BC, DE

are equal, the chords

AC, BD, CE

are all equal; let

x = AC = BD = CE,

y = AD,

and

z = BE.

Ptolemy’s theorem on

ABCD,

BCDE,

and

ABDE

gives

\begin{gathered} 10y + 9 = x^2, \\ \quad 100 + 3z = x^2, \\ \quad 30 + 14x = yz. \end{gathered}

Solving the first two for

y

and

z

and substituting into the third yields

\begin{gathered} x^3 - 109x - 420 = 0 \\ = (x-12)(x+5)(x+7). \end{gathered}

So

x = 12,

y = \tfrac{135}{10} = \tfrac{27}{2},

and

z = \tfrac{44}{3}.

The five diagonals are

AC, BD, CE, AD, BE,

summing to

\begin{gathered} 3x + y + z = 36 \\ {}+ \tfrac{27}{2} + \tfrac{44}{3} \\ = \tfrac{385}{6}. \end{gathered}

Thus

m + n = 385 + 6 = 391,

and the correct answer is D.

其他年份的第 24 题