1956 AMC 12 第 43 题

Question

三边长均为整数且周长小于

13

的不等边三角形共有：

The number of scalene triangles having all sides of integral lengths, and perimeter less than

13

is:

1

2

3

4

18

将互不相同的整数边长按

a\lt b\lt c

排列，并利用

a+b\gt c

Order the distinct integer sides as

a\lt b\lt c

and use

a+b\gt c

按最大边逐一列出可能；周长限制使

c

只能取较小的值

List possibilities by the largest side; the perimeter bound leaves only small values of

c

将互不相同的整数边长按递增顺序排列。在三角形不等式和周长限制下检查较小的可能，得到

\begin{aligned} &(2,3,4),\ (2,4,5),\\ &(3,4,5) \end{aligned}\text{。}

当

c\ge6

时，满足

a+b\gt c

的最小新不等边候选为

(3,4,6)

，其周长已经是

13

，更大的选择都不合要求。因此共有

3

个三角形。

正确答案是 C。

Write the distinct integer sides in increasing order. Checking the small possibilities under the triangle inequality and perimeter bound gives

\begin{aligned} &(2,3,4),\ (2,4,5),\\ &(3,4,5). \end{aligned}

For

c\ge6,

the smallest new scalene candidate satisfying

a+b\gt c

is

(3,4,6),

whose perimeter is already

13,

and larger choices cannot qualify. Thus there are

3

triangles.

The correct answer is C.

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