# 《毕出21套》解析：数学第1套

## 数学第1套PS

1.How many positive integers n have the property that both 3n and n/3 are 4-digit integers?

(A) 111

(B) 112

(C) 333

(D) 334

(E) 1,134

### 解：

2.If Whitmey wrote the decimal representations for the first 300 positive integer multiples of 5 and did not write any other numbers, how many times would she have written the digit 5?

(A) 150

(B) 185

(C) 186

(D) 200

(E) 201

### 解：

3.If x < y < z and y - x > 5, where x is an even integer and y and z are odd integers, what is the least possible value of z-x?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

### 解：

zy都是奇数，所以z最小为y+2，即，x+7+2。由此可知，z-x最小为9，答案为D。

4.If the product of the integers w, x, y, and z is 770, and if 1 < w < x < y < z, what is the value of w + z?

(A) 10

(B) 13

(C) 16

(D) 18

(E) 21

### 解：

5.An "Armstrong number" is an n-digit number that is equal to the sum of the nth powers of its individual digits. For example, 153 is an Armstrong number because it has 3 digits and 13+53+33=153. What is the digit k in the Armstrong number 1,6, k, 4？

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

### 解：

14+64+k4+44=1604+10*k （这里最关键的就是把16k4变成1604+10*k ）

k(k3-10) = 1604-1553=51

6.If n=p2 and p is a prime number greater than 5, what is the units digit of n2?

(A) 1

(B) 3

(C) 4

(D) 7

(E) 9

### 解：

7.Let S be the set of all positive integers having at most 4 digits and such that each of the digits is 0 or 1. What is the greatest prime factor of the sum of all the numbers in S?

(A) 11

(B) 19

(C) 37

(D) 59

(E) 101

### 解：

0，1，10，11，1000，1001，1010，1011，1100，1101，1110，1111

8.When the integer n is divided by 17, the quotient is x and the remainder is 5. When n is divided by 23, the quotient is y and the remainder is 14. Which of the following is true?

(A) 23x+17y=19

(B) 17x-23y=9

(C) 17x+23y=19

(D) 14x+5y=6

(E) 5x-14y=-6

### 解：

n = 17x + 5 = 23y + 14，由此可解出17x-23y=9，答案为B。

9. 1,234

1,243

1,324

........

........

+4,321

The addition problem above shows four of the 24 different integers that can be formed by using each of the digits 1, 2, 3, and 4 exactly once in each integer. What is the sum of these 24 integers?

(A) 24,000

(B) 26,664

(C) 40,440

(D) 60,000

(E) 66,660

### 解：

60*(103 + 102 + 10 + 1) = 66660，答案为E。

10.A computer can perform 1,000,000 calculations per second. At this rate, how many hours will it take this computer to perform the 3.6 x 1011 calculations required to solve a certain problems?

(A) 60

(B) 100

(C) 600

(D) 1,000

(E) 6,000

### 解：

11.A school supply store sells only one kind of desk and one kind of chair, at a uniform cost per desk or per chair. If the total cost of 3 desks and 1 chair is twice that of 1 desk and 3 chairs, then the total cost of 4 desks and 1 chair is how many times that of 1 desk and 4 chairs?

(A) 5

(B) 3

(C) 8/3

(D) 5/2

(E) 7/3

### 解：

3d+c=2d+6c

d=5c

12. Let n and k be positive integers with k ≤ n. From an n × n array of dots, a k × k array of dots is selected. The figure above shows two examples where the selected k × k array is enclosed in a square. How many pairs (n, k) are possible so that exactly 48 of the dots in the n × n array are NOT in the selected k × k array?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5

### 解：

n*n - k*k = 48

(n - k) * (n + k) = 48

(2，24), (4，12), (8，6), (16，3)

13.A car traveled 462 miles per tankful of gasoline on the highway and 336 miles per tankful of gasoline in the city. If the car traveled 6 fewer miles per gallon in the city than on the highway, how many miles per gallon did the car travel in the city?

(A) 14

(B) 16

(C) 21

(D) 22

(E) 27

### 解：

14. A certain manufacturer uses the function C(x)=0.04x2-8.5x+25,000 to calculate the cost, in dollars, of producing x thousand units of its product. The table above gives values of this cost function for values of x between 0 and 50 in increments of 10. For which of the following intervals is the average rate of decrease in cost less than the average rate of decrease in cost for each of the other intervals?

(A) From x=0 to x=10

(B) From x=10 to x=20

(C) From x=20 to x=30

(D) From x=30 to x=40

(E) From x=40 to x=50

### 解：

15.There were 36, 000 hardback copies of a certain novel sold before the paperback version was issued. From the time the first paperback copy was sold until the last copy of the novel was sold, 9 times as many paperback copies as hardback copies were sold. If a total of 441,000 copies of the novel were sold in all, how many paperback copies were sold?

(A) 45,000

(B) 360,000

(C) 364,500

(D) 392,000

(E) 396,900

### 解：

X + Y = 441000 - 36000 = 405000

9X = Y

Y = 364500，答案为C。

16.A certain truck traveling at 55 miles per hour gets 4.5 miles per gallon of diesel fuel consumed. Traveling at 60 miles per hour, the truck gets only 3.5 miles per gallon. On a 500-mile trip, if the truck used a total of 120 gallons of diesel fuel and traveled part of the trip at 55 miles per hour and the rest at 60 miles per hour, how many miles did it travel at 55 miles per hour?

(A) 140

(B) 200

(C) 250

(D) 300

(E) 360

### 解：

X/4.5 + (500 - X)/3.5 = 120

17.A merchant paid \$300 for a shipment of x identical calculators. The merchant used two of the calculators as demonstrators and sold each of the others for \$5 more than the average(arithmetic mean) cost of the x calculators. If the total revenue from the sale of the calculators was \$120 more than the cost of the shipment, how many calculators were in the shipment?

(A) 24

(B) 25

(C) 26

(D) 28

(E) 30

### 解：

(300/x + 5)(x-2) = 420

18.If there is a least integer that satisfies the inequality 9/x ≥2, what is that least integer?

(A) 0

(B) 1

(C) 4

(D) 5

(E) There is not a least integer that satisfies the inequality

### 解：

19.The sum of the ages of Doris and Fred is y years. If Doris is 12 years older than Fred, how many years old will Fred be y years from now, in terms of y?

(A) y-6

(B) 2y-6

(C) y/2-6

(D) 3y/2-6

(E) 5y/2-6

### 解：

D + F = y

D - F = 12

F = y/2 - 6

F + y =3y/2 – 6，答案为D。

20.If k≠0 and k- (3−2k2)/k = x/k, then x=

(A) -3-k2

(B) k2-3

(C) 3k2-3

(D) k-3-2k2

(E) k-3+2k2

### 解：

21.King School has an enrollment of 900 students. The school day consists of 6 class periods during which each class is taught by one teacher. There are 30 students per class. Each teacher teaches a class during 5 of the 6 class periods and has one class period free. No students have a free class period. How many teachers does the school have?

(A) 25

(B) 30

(C) 36

(D) 60

(E) 150

### 解：

900个学生，6个课时，1个老师教1个班，每个班30个学生，每个老师教5个课时休息1个课时。

900/30=30个班

30*6=180课时

180/5=36个老师，答案为C。

22.Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10？

(A) Three

(B) Four

(C) Five

(D) Six

(E) Seven

### 解：

1+9

2+8

3+7

4+6

0

5

23. The figure above represents a network of one-way streets. The arrows indicate the direction of traffic flow and the numbers indicate the amount of traffic flow into or out of each of the four intersections during a certain hour. During that hour, what was the amount of traffic flow along the street from R to S if the total amount of traffic flow into P was 1, 200?

(Assume that none of the traffic originates or terminates in the network)

(A) 200

(B) 250

(C) 300

(D) 350

(E) 400

### 解：

P是1200， 流入P点的是800和S，流入S的是550和R，但是S还要流出400。

1200-800+400（加回S流出的400）-550（减去流入的550）=250（PS），答案为B。

24.Machines X and Y run at different constant rates, and machine X can complete a certain job in 9 hours. Machine X worked on the job alone for the first 3 hours and the two machine working together, then completed the job in 4 more hours. How many hours would it have taken machine Y, working alone, to complete the entire job?

(A) 18

(B) 13+1/2

(C) 7+1/5

(D) 4+1/2

(E) 3+2/3

### 解：

X的速率为 1/9，Y的速率为 1/T

7*1/9 + 4*1/T = 1

25. When a certain stretch of highway was rebuilt and straightened, the distance along stretch was decreased by 20 percent and the speed limit was increased by 25 percent. By what percent was the driving time along this stretch reduced for a person who always drives at the speed limit?

(A) 16%

(B) 36%

(C) 37.5%

(D) 45%

(E) 50.25%

### 解：

26.The annual stockholders' report for Corporation X stated that profits were up 10 percent over the previous year, although profits as a percent of sales were down 10 percent. Total sales for that year were approximately what percent of sales for the previous year?

(A) 78%

(B) 90%

(C) 110%

(D) 122%

(E) 190%

### 解：

1.1*P1/ S2 = 0.9*P1/S1

S2= 1.1*S1/0.9= 1.222*S1，答案为D。

27.A certain brand of house paint must be purchased either in quarts at \$12 each or in gallons at \$18 each. A painter needs a 3-gallon mixture of the paint consisting of 3 parts blue and 2 parts white. What is the least amount of money needed to purchase sufficient quantities of the two colors to make the mixture?

(4 quarts= 1 gallon)

(A) \$54

(B) \$60

(C) \$66

(D) \$90

(E) \$144

### 解：

28. If each side of △ACD above has length 3 and if AB has length 1, what is the area of region BCDE?

(A) 9/4

(B) (7/4)√3

(C) (9/4)√3

(D) (7/2)√3

(E) 6+ √2

### 解：

29. The figure above shows some of the dimensions of a triangular plaza with an L-shaped walk along two of its edges. If the width of the walk is 4 feet and the total area of the plaza and walk together is 10,800 square feet, what is the value of x?

(A) 200

(B) 204

(C) 212

(D) 216

(E) 225

### 解： 30. The surface distance between 2 points on the surface of a cube is the length of the shortest path on the surface of the cube that joins the 2 points. If a cube has edges of length 4 centimeters, what is the surface distance, in centimeters, between the lower left vertex on its front face and the upper right vertex on its back face?

(A) 8

(B) 4 √5

(C) 8√2

(D) 12√2

(E) 4√2+4

### 解：

31. In the figure above, the area of the shaded region is

(A) 8√2

(B) 4√3

(C) 4√2

(D) 8(√3-1)

(E) 8(√2-1)

### 解：

32. In the figure above, PQ is a diameter of circle O, PR=SQ, and △RST is equilateral. If the length of PQ is 2, what is the length of RT?

(A) 1/2

(B) 1/√3

(C) √3/2

(D) 2/√3

(E) √3

### 解：

33.A circular rim 28 inches in diameter rotates the same number of inches per second as a circular rim 35 inches in diameter. If the smaller rim makes x revolutions per second, how many revolutions per minute does the larger rim make in terms of x?

(A) 48π/x

(B) 75x

(C) 48x

(D) 24x

(E) x/75

### 解：

2π*14*x=2π*17.5*y

34. Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate tums, and no stick that is removed is put back on the table. The one who removes the last stick(or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays？

(A) 7

(B) 10

(C) 11

(D) 12

(E) 16

### 解：

35. Ben and Ann are among 7 contestants from which 4 semifinalists are to be selected. Of the different possible selections, how many contain neither Ben nor Ann?

(A) 5

(B) 6

(C) 7

(D) 14

(E) 21

### 解：

36. The map above shows the trails through a wilderness area. If travel is in the direction of the arrows, how many routes along the marked trails are possible from point A to point B?

(A) 11

(B) 18

(C) 54

(D) 108

(E) 432

### 解：

27+27=54，答案为C。

37. Pat will walk from intersection X to intersection Y along a route that is confined to the square grid of four streets and three avenues shown in the map above. How many routes from X to Y can Pat take that have the minimum possible length?

(A) Six

(B) Eight

(C) Ten

(D) Fourteen

(E) Sixteen

### 解：

38. If s/t =2, then the value of which of the following can be determined?

I 2t/s

II (s−t)/t

III (t−1)/(s−1)

(A) I only

(B) III only

(C) I and II only

(D) II and III only

(E) I and II and III

### 解：

39. If 2x+5y=8 and 3x=2y, what is the value of 2x+y?

(A) 4

(B) 70/19

(C) 64/19

(D) 56/19

(E) 40/19

### 解：

40. When 2/9 of the votes on a certain resolution have been counted, 3/4 of those counted are in favor of the resolution. What fraction of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution?

(A) 11/14

(B) 13/18

(C) 4/7

(D) 3/7

(E) 3/14

### 解：

41.The sum of the first 100 positive integers is 5,050. What is the sum of the first 200 positive integers?

(A) 10,100

(B) 10,200

(C) 15,050

(D) 20,050

(E) 20,100

### 解：

101到200的和相当于1到100的和后面每个数都加上100，5050*2 + 100*100 = 20100，答案为E。