# 美国数学邀请赛2007AMC10试题

8th AMC 10 A 8th AMC 10 A
Instructions

2007 2007

1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 2. You will receive 6 points for each correct answer, 1.5 points for each problem left unanswered, and 0 points for each incorrect answer. 3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (No problems on the test will require the use of a calculator). 4. Figures are not necessarily drawn to scale. 5. You will have 75 minutes working time to complete the test

1. One ticket to a show costs \$20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? (A) 2 (B) 5 (C) 10 (D) 15 (E) 20 中文：一张展览票全价为 20 美元。Susan 用优惠券买 4 张票打七五折。Pam 用优惠券 买 5 张票打七折。Pam 比 Susan 多花了多少美元？C 2. Define a@b=ab - b2 and a#b=a + b - ab2. 1 (A)－ 2 1 (B)－ 4 (C) 1 8 (D) 1 4 6@2 ? 6#2 1 2

What is

(E)

6@2 中文：定义 a @ b=ab-b2 ，a # b=a + b - ab2。求 的值。A 6#2 3. An aquarium has a rectangular base that measures 100cm by 40cm and has a height of 50cm. It is filled with water to a height of 40cm. A brick with a rectangular base that measures 40cm by 20cm and a height of 10cm is placed in the aquarium. By how many centimeters does that water rise? (A) 0.5 (B) 1 (C) 1.5 (D) 2 (E)2.5 中文：一个养鱼缸有 100cm×40cm 的底，高为 50cm。它装满水到 40cm 的高度。把一 个底为 40cm×20cm，高为 10cm 的砖块放在这个养鱼缸里。鱼缸里的水上升了多少厘 米？D 4. The larger of two consecutive odd integers is three times the smaller. What is their sum?

8th AMC 10 A

2007

(A) 4 (B) 8 (C) 12 (D) 16 (E) 20 中文：2 个连续的奇整数中较大的数是较小的数的 3 倍。它们的和是多少？A 5. A school store sells 7 pencils and 8 notebooks for \$4.15. It also sells 5 pencils and 3 notebooks for \$1.77. How much do 16 pencils and 10 notebooks cost? (A) \$4.76 (B) \$5.84 (C) \$6.00 (D) \$6.16 (E) \$6.32 中文：在一个学校商店花 4.15 美元可以买到 7 支铅笔和 8 个笔记本，花 1.77 美元可以 买到 5 个铅笔和 3 个笔记本。问 16 支铅笔和 10 个笔记本需要多少钱？B 6. At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, and 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? (A) 2002 and 2003 (B) 2003 and 2004 (C) 2004 and 2005 (D) 2005 and 2006 (E) 2006 and 2007 中文：在 Euclid 高中，2002 年参加 AMC10 的学生有 60 人，2003 年有 66 人，2004 年 有 70 人，2005 年有 76 人，2006 年有 78 人，2007 年有 85 人。在相邻的哪两年中人数 增长的百分率最大？A 7. Last year Mr. John Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state taxes. He paid a total of \$10,500 for both taxes. How many dollars was the inheritance? (A) 30,000 (B) 32,500 (C) 35,000 (D) 37,500 (E) 40,000 中文：去年 Mr. John Q. Public 收到了一笔遗产。其中 20%用于交联邦税，剩下钱中的 10%用于交州税。交这两种税款一共需要 10,500 美元。问这笔遗产共多少美元？D 8. Triangles ABC and ADC are isosceles with AB=AC and AD=DC. Point D is inside △ABC, ∠ABC=40°, and ∠ADC=140°. What is the degree measure of ∠BAD? (A) 20 (B) 30 (C) 40 (D) 50 (E) 60 中文：△ABC 和△ADC 是等腰三角形且 AB=AC，AD=DC。点 D 在△ABC 内，∠ ABC=40°, ∠ADC=140°。问∠BAD 的度数是多少？(此题似题目出错) 9. Real numbers a and b satisfy the equations 3 =81 and 125 =5 . What is ab? (A)－60 (B)－17 (C) 9 (D) 12 (E) 60 a b+2 b a-3 中文：实数 a 和实数 b 满足等式 3 =81 和 125 =5 ，问 ab 的值是多少？E 10. The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20, the father is 48 years old, and the average age of the mother and children is 16. How many children are in the family?

a b+2 b a-3

8th AMC 10 A

2007

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6 中文：Dunbar 一家的家庭成员包括爸爸，妈妈和几个孩子。他们的平均年龄是 20 岁， 其中爸爸 48 岁，妈妈和几个孩子的平均年龄是 16 岁。问家中有多少个孩子？E 11. The numbers from 1 to 8 are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? (A) 14 (B) 16 (C) 18 (D) 20 (E) 24 中文：数字 1 到 8 被放在立方体的 8 个顶点上。使每面上的四个顶点代表的数字和相 同。问每面上的四个顶点的数字和是多少？C 12. Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible? (A) 56 (B) 58 (C) 60 (D) 62 (E) 64 中文：两位导游带领六位游客旅游，导游决定分开带路，每位游客必须选择一位导游， 且规定只能选一位导游，又每位导游至少要带领一位游客，由导游和游客组成不同团 体总共有多少种可能组合？D 13. Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan’s distance from his home to his distance from the stadium? (A) 2 3 (B) 3 4 (C) 4 5 (D) 5 6 (E) 6 7

3

8th AMC 10 A

2007

(A) 32

(B) 22+12 2

(C) 16+16 3

(D) 48

(E) 36+16 2

3 8

(B)

7 16

(C)

1 2

(D)

9 16

(E)

5 8

3 3

44 3

(B) 16

(C)

88 5

(D) 20

(E)

62 3

8th AMC 10 A
the square to the brush width?

2007

(A) 2 2 +1

(B) 3 2

(C) 2 2 +2

(D) 3 2 +1

(E) 3 2 +2

-1 4 -4 -1 4 -4

8th AMC 10 A

2007

(A) 3 (B) 4 (C) 6 (D) 9 (E)12 中文：有多少个正整数对（m, n）满足 m > n 且它们的平方差是 96？A 24. Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of AB , and OA=2 2 . Segments OC and OD are tangent to the circles centered at A and B, respectively, and EF is a common tangent. What is the area of the shaded region ECODF?

(A)

3 8

2

(B)8 2 －4－π

(C) 4 2

(D)4 2 +

π 8

(E)8 2 －2－

π 2

6

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