AMC10 2003 A

AMC10 2003 A · Q23

AMC10 2003 A · Q23. It mainly tests Arithmetic sequences basics, Polygons.

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 3 rows of small congruent equilateral triangles, with 5 small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 2003 small equilateral triangles?

一个大正三角形由牙签构成小正三角形的行来构建。例如，图中 3 行小正三角形，底行有 5 个小三角形。如果底行由 2003 个小正三角形构成，需要多少牙签来构建这个大正三角形？

(A) 1,004,004 1,004,004

(B) 1,005,006 1,005,006

(D) 3,015,018 3,015,018

(E) 6,021,018 6,021,018

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) The base row of the large equilateral triangle has 1001 triangles pointing downward and 1002 pointing upward. This base row requires $3(1002)$ toothpicks since the downward pointing triangles require no additional toothpicks. Each succeeding row will require one less set of $3$ toothpicks, so the total number of toothpicks required is $3(1002 + 1001 + 1000 + \cdots + 2 + 1)=3\cdot\frac{1002\cdot1003}{2}=1,507,509.$

（C）这个大等边三角形的底边一行有 1001 个倒三角形和 1002 个正三角形。由于倒三角形不需要额外的牙签，这一行需要 $3(1002)$ 根牙签。之后每一行所需的牙签组数都会少一组 $3$ 根牙签，因此所需牙签总数为 $3(1002 + 1001 + 1000 + \cdots + 2 + 1)=3\cdot\frac{1002\cdot1003}{2}=1,507,509.$

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