2014 AMC 12B 第 12 题

Question

集合

S

由边长均为小于

5

的正整数的三角形组成，且

S

中没有两个元素全等或相似。

S

最多可以有多少个元素？

A set

S

consists of triangles whose sides have integer lengths less than

5,

and no two elements of

S

are congruent or similar. What is the largest number of elements that

S

can have?

8

9

10

11

12

用

\{1,2,3,4\}

中的数按非递增顺序列出每个三角形的三边。

List each triangle by its sides in nonincreasing order using values from

\{1,2,3,4\}

去掉不满足三角形不等式的，并注意

2,2,1

和

4,4,2

相似，只能计一次。

Drop those failing the triangle inequality, and count

2,2,1

and

4,4,2

only once since they are similar

按非递增顺序写出每个三角形的边长。等边三角形只能选一个（它们都相似），并且相似的一对

2,2,1

和

4,4,2

中也只能选一个。

其余有效且两两不相似的三角形为

\begin{gathered} 4\,4\,3,\ 4\,4\,1,\ 4\,3\,3,\ 4\,3\,2,\ \\ 3\,3\,2,\ 3\,3\,1,\ 3\,2\,2\text{，} \end{gathered}

一共七个。再加上一个等边三角形和相似对中的一个，

S

最多有

9

个元素。

所以正确答案是 B。

Write each triangle by its side lengths in nonincreasing order. Only one equilateral triangle is allowed (all are similar), and of the similar pair

2,2,1

and

4,4,2

only one may appear.

The remaining valid, pairwise non-similar triangles are

\begin{gathered} 4\,4\,3,\ 4\,4\,1,\ 4\,3\,3,\ 4\,3\,2,\ \\ 3\,3\,2,\ 3\,3\,1,\ 3\,2\,2, \end{gathered}

seven in all. Together with one equilateral and one of the similar pair,

S

has at most

9

elements.

Thus, the correct answer is B.

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