AMC12 2014 B
AMC12 2014 B · Q12
AMC12 2014 B · Q12. It mainly tests Basic counting (rules of product/sum), Triangles (properties).
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?
集合$S$由边长小于5的整数长度的三角形组成，且$S$中没有两个元素全等或相似。$S$的最大元素个数是多少？
(A)
8
8
(B)
9
9
(C)
10
10
(D)
11
11
(E)
12
12
Answer
Correct choice: (B)
正确答案：（B）
Solution
Answer (B): Denote a triangle by the string of its side lengths written in nonincreasing order. Then $S$ has at most one equilateral triangle and at most one of the two triangles 442 and 221. The other possible elements of $S$ are 443, 441, 433, 432, 332, 331, and 322. All other strings are excluded by the triangle inequality. Therefore $S$ has at most 9 elements.
答案（B）：用按非增顺序排列的边长所组成的字符串来表示一个三角形。则集合 $S$ 至多包含一个等边三角形，并且在三角形 442 与 221 这两个之中至多包含一个。$S$ 的其他可能元素为 443、441、433、432、332、331 和 322。其余所有字符串都由三角形不等式排除。因此，$S$ 至多有 9 个元素。
Topics
Related Questions
Practice full AMC exams on amcdrill.
Try full-length practice and diagnostics at www.amcdrill.com.