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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?
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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?
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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

中文:Yan 站在他家和运动场之间,要到运动场。他可以直接走路到运动场,或他走路 回家再沿原路径骑自行车到运动场。骑自行车的速率是走路速率的 7 倍,两种方法所 花的时间都相同。问 Yan 站的位置到他家的距离与到运动场的距离之比值是多少?B 14. A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? (A) 8.64 (B) 12 (C) 5π (D) 17.28 (E) 18 中文:一个三角形的外接圆半径是 3,且其三边长之比是 3:4:5。问此三角形的面积是 多少?A 15. Four circles of radius 1 are each tangent to two sides of a square and externally tangent to a circle of radius 2, as shown. What is the area of the square?

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3



8th AMC 10 A

2007

(A) 32

(B) 22+12 2

(C) 16+16 3

(D) 48

(E) 36+16 2

中文:如图所示,四个半径是 1 的圆分别与正方形的两边相切且与一个半径是 2 的圆 外切,这个正方形的面积是多少?B 16. Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that ad - bc is even? (A)

3 8

(B)

7 16

(C)

1 2

(D)

9 16

(E)

5 8

中文:从 0 到 2007 中任意地选出整数 a,b,c,d,它们不必都相异,问 ad-bc 为偶数的 几率是多少?E 17.Suppose that m and n are positive integers such that 75m=n . What is the minimum possible value of m + n? (A) 15 (B) 30 (C) 50 (D) 60 (E) 5700 中文:设 m 和 n 为正整数且满足 75m=n 。问 m + n 所有可能值中的最小值是多少?D 18. Consider the 12-sided polygon ABCDEFGHIJKL, as shown. Each of its sides has length 4, and each two consecutive sides form a right angle. Suppose that AG and CH meet at M. What is the area of quadrilateral ABCM? 0000 (A)
3 3

44 3

(B) 16

(C)

88 5

(D) 20

(E)

62 3

中文:如图所示,十二边形 ABCDEFGHIJKL 中每边的边长都是 4,每两邻边都互相 垂直,设 AG 与 CH 交于 M 点,问四边形 ABCM 的面积是多少?(图为空心十字形, A 位于左上角,顺时针排列)C 19.A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of
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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

中文:如图所示,沿着正方形的对角线以油漆刷子刷出两个对称于对角线的全等涂色 带状区域。如果图中涂色区域的面积是正方形面积的一半,那么正方形边长与油漆刷 子的宽度的比值是多少?C 20.Suppose that the number a satisfies the equation 4 = a + a . What is the value of a + a ? (A) 164 (B) 172 (C) 192 (D) 194 (E) 212 中文:设数 a 满足等式 4 = a + a ,问 a +a 的值是多少?D 21. A sphere is inscribed in a cube that has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? (A) 3 (B) 6 (C) 8 (D) 9 (E) 12 中文:一个表面积为 24 平方米的正立方体内有一个内切球,且球内又有一个内接正立 方体,问球内的正立方体的表面积是多少平方米?E 22.A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with terms 247,475, and 756 and end with the term 824. Let S be the sum of all the terms in the sequence. What is the largest prime number that always divides S? (A) 3 (B) 7 (C) 13 (D) 37 (E) 43 中文:考虑有限数列,每项均为三位数,且具有下列性质:每一项十位数字与个位数 字分别是下一项的百位数字与十位数字,最后一项的十位数字与个位数字是第一项的 百位数字与个位数字。例如,247,475,756,…,824 就是一个这样的数列。用 S 表示这种数列各项的和。下面哪一个数是一定可以整除 S 的最大质数?A 23. How many ordered pairs (m, n) of positive integers, with m > n, have the property that their square differ by 96?
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-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

中文:如图所示, 以 A, B 为圆心、以 2 为半径的两个圆。点 O 是 AB 的中点,且 OA=2 2 ,线段 OC 与 OD 分别切⊙A 及⊙B 于 C, D 两点,且 EF 是两圆的公切线, 问阴影区域 ECODF 的面积是多少?B 25.For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n + S(n) + S(S(n)) = 2007? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 中文:对每一个正整数 n,以 S(n)表示 n 各位数字之和,问使得 n + S(n) + S(S(n))=2007 的 n 有多少个?D

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