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2014-2015年度美国”数学大联盟杯赛“（中国赛区）初赛

（十、十一、十二年级）

一、选择题（每小题10分，答对加10分，答错不扣分，共100分，请将正确答案A、B、C或者D写在每题后面的圆括号内。）

正确答案填写示例如下： 20?5?2=2? ？ （A）

A)5 B)15

C)25 D)30

1. Meg loves her megaphone! The large circular end has a circumference that is the reciprocal of its diameter. What is the area of the circle? ( )

A)

1 4?B)

1 2?C)

1 4D)

1 22. How many solutions does the equation 2x?3?3x have? ( )

A)0

B)1

C)2

D)4

3. If y?1?x, which of the following is always true for any value of x? ( )

A) (1?x)?(1?y) C) x?(1?x)?y(1?y)

222222B) (1?x)x?(1?y)y

D) (1?x)(1?x)?(1?y)(1?y)

222222224. Lee the crow ate a grams of feed that was 1% seed, b grams of feed that was 2% seed, and c grams of feed that was 3% seed. If combined, all the feed he ate was 1.5% seed. What is a in terms of b and c ? ( ) A)b?3c B)3b?c

C)2b?3c

D)3b?2c

2?15. If x<0 and x<0.01, then x must be ( )

A)less than -10 C)between 0and 0.1

B)between-0.1 and 0 D) greater than 10

6. At 9:00 A.M., the ratio of red to black cars in a parking lot was 1 to 5. An hour later the number of red cars had increased by 2, the number of black cars had decreased by 5, and the ratio of red to black cars was 1 to 4. How many black cars were in the lot at 10:00 A.M.? ( ) A)13 B)15 C)60 D)65

4x3(x2?1)?x2?17. If x?1and x??1, then =( )

(x?1)(x?1)A)x?1

2B) x?1

2C) 4x?1

2D) 4x?1

38. The Camps are driving at a constant rate. At noon they had driven 300 km. At 3:30 P.M. they had driven 50% further than they had driven by 1:30 P.M. What is their constant rate in km/hr? ( )

A)150 B)120 C)100 D)90

9. The letters in DIGITS can be arranged in how many orders without adjacent I’s? ( )

A)240 B)355 C)600 D)715 10. Al, Bea, and Cal each paint at constant rates, and together they are painting a house. Al and Bea together could do the job in 12 hours; Al and Cal could to it in 15, and Bea and Cal could do it in 20. How many hours will it take all three working together to paint the house?( ) A)8.5 B)9 C)10 D)10.5

二、填空题（每小题10分，答对加10分，答错不扣分，共200分）

11. What is the sum of the degree-measures of the angles at the outer points

BCA,B,C,D and E of a five-pointed star, as shown?

Answer: .

AED12. What is the ordered pair of positive integers (k,b), with the least value of k, which satisfies

2?33?44?kb? Answer: .

13. A face-down stack of 8 playing cards consisted of 4 Aces (A’s) and 4 Kings (K’s). After I revealed and then removed the top card, I moved the new top card to the bottom of the stack without revealing the card. I repeated this procedure until the stack without revealing the card. I repeated this procedure until the stack was left with only 1 card, which I then revealed. The cards revealed were AKAKAKAK, in that order. If my original stack of 8 cards had simply been revealed one card at a time, from top to bottom (without ever moving cards to the bottom of the stack), in what order would they have been revealed? Answer: . 14. For what value of a is one root of x?(2a?1)x?a?2?0 twice the other root? Answer: .

15. Each time I withdrew $32 from my magical bank account, the account’s remaining balance doubled. No other account activity was permitted. My fifth

22$32 withdrawal caused my account’s balance to become $0. With how many

dollars did I open that account? Answer: .

16. In how many ways can I select six of the first 20 positive integers, disregarding the order in which these six integers are selected, so that no two of the selected integers are consecutive integers?

Answer: . 17. If, for all real x,f(x)?2f(1?x), what is the numerical value of f(3)?

Answer: .

18. How many pairs of positive integers (without regard to order) have a least common multiple of 540?

Answer: .

x19. If the square of the smaller of consecutive positive integers is x, what is the square of the larger of these two integers, in terms of x? Answer: .

20. A pair of salt and pepper shakers comes in two types: identical and fraternal. Identical pairs are always the same color. Fraternal pairs are the same color half the time. The probability that a pair of shakers is fraternal is p and that a pair is identical is q?1?p. If a pair of shakers is of the same color, determine, in terms of the variable q alone, the probability that the pair is identical. Answer: .

21. As shown, one angle of a triangle is divided into four smaller congruent

angles. If the lengths of the sides of this triangle are 84, 98, and 112, as shown, how long is the segment marked x ?

Answer: .

22. How long is the longer diagonal of a rhombus whose perimeter is 60, if three of its vertices lie on a circle whose diameter is 25, as shown? Answer: .

23. The 14 cabins of the Titanic Mail Boat are numbered consecutively from 1 through 14, as are the 14 room keys. In how many different ways can the 14 room keys be placed in the 14 rooms, 1 per room, so that, for every room, the sum of that room’s number and the number of the key placed in that room is a multiple of 3? Answer: . 24. For some constant b, if the minimum value of

8411298xx2?2x?b1f(x)?2is , what is the maximum value of f(x)?

2x?2x?bAnswer: .

25. If the lengths of two sides of a triangle are 60cos A and 25sin A, what is the greatest

possible integer-length of the third side? Answer: . 26. ?an?is a geometric sequence in which each term is a positive number. If a5a6?27, what is

the value of log3a1?log3a2?K?log3a10? Answer: .

27. What is the greatest possible value of f(x)=3sinx?4cosx?12? Answer: .

28. Let C be a cube. Triangle T is formed by connecting the midpoints of three edges of cube

C. What is the greatest possible measure of an angle of triangle T? Answer: . 29. Let a and b be two real numbers. f(x)?asinx?b3x?4 and f(lglog310)?5.

What is the value of f(lglg3)?

Answer: .

30. Mike likes to gamble. He always bets all his chips whenever the number of chips he has is

<=5. He always bets （10-n）chips whenever the number of chips he has is greater than 5

and less than 10. He continues betting until either he has no chips or he has more than 9 chips. For every round, if he bets n chips. The probability that he wins or loses in each round is 50%. If Mike begins with 4 chips, what is the probability that he loses all his chips? Answer: .