MATHCOUNTS

2001 Chapter Competition

Countdown Round

1. Three coins are flipped. What is the probability that all three coins land heads? Express your answer as a common fraction.

Answer: 1

8

2. How many positive integers have exactly three digits?

Answer: 900 (integers)

3. A bookstore employee is given a 30% discount off the retail price of any book. How many dollars does she pay for a book with a retail price of $40?

Answer: 28 (dollars)

4. How many prime numbers are between 20 and 30?

Answer: 2 (numbers)

5. The number 12 can be written as the sum of eight consecutive integers. What is the product of these integers?

Answer: 0

6. Fifteen is twenty percent of what number?

Answer: 75

7. When Monique visited Australia, the American dollar was equivalent to 1 Australian dollars. She purchased a sweater priced at 30 Australian dollars. What was the equivalent number of American dollars?

12

Answer: 20 (dollars)

8. What is the greatest common factor of 32 and 48?

Answer: 16

9. What is the sum of the mean and the median of the set {15, 17, 19, 21, 23} ?

Answer: 38

10. If x is 125% of y, then what percent of x is 2y?

Answer: 160 (percent)

11. When all two-digit positive integers are written, what fraction of the digits written are 3’s? Express your answer as a common fraction.

Answer: 19

180

12. License plates are issued that contain four digits followed by one letter. If the letters O and I cannot be used, how many different license plates are possible?

Answer: 240,000 (plates)

13. A bus can hold a maximum of 38 students. What is the minimum number of buses needed to transport 411 students?

Answer: 11 (buses)

14. How many terms does the arithmetic sequence 15, 20, 25, 30, ..., 95 have?

Answer: 17 (terms)

15. J.C. rode her bike for 1 hours at 12 miles per hour, and then rode for 1 hours at 8 miles per hour. What is the total number of miles she rode?

12

34

Answer: 33 (miles)

16. What is the number of square centimeters in the surface area of a rectangular prism whose length is 3 cm, whose width is 4 cm, and whose height is 5 cm?

4

53

Answer: 94 (square centimeters)

17. Find the integer closest

to .7003

Answer: 9

18. Twenty-five percent of the entrants in a marathon did not finish the race. If 375 finished, how many entrants were there?

Answer: 500 (entrants)

19. How many different two-letter monograms are possible if letters may be repeated?

Answer: 676 (monograms)

20. The product of two consecutive odd whole numbers is 255. What is the greater number?

Answer: 17

21. For a concert, 72 fans attempted to get tickets. That was 60% more than the number of tickets available. How many tickets were available?

Answer: 45 (tickets)

22. How many diagonals does a regular hexagon have?

Answer: 9 (diagonals)

23. If the price of a stamp is 33¢, what is the maximum number of stamps that could be purchased with $32?

Answer: 96 (stamps)

24. Set A contains 15 elements, set B contains 12 elements, and the intersection of the sets contains 8 elements. How many elements are in the union of the sets?

Answer: 19 (elements)

25. What is the number of edges in a regular hexagonal prism?

Answer: 18 (edges)

26. A racer travels 5000 meters in 1 minutes. What is the racer’s average speed in kilometers per hour?

23

Answer: 180 (kilometers per hour)

27. A plane is flying at an altitude of 22,000 feet when it begins to descend at a constant rate of 2000 feet per minute. What is its altitude, in feet, after 75 seconds?

Answer: 19,500 (feet)

28. A woman spent $57 on a skirt and a sweater. If the sweater cost twice as much as the skirt, how many dollars were in the price of the skirt?

Answer: 19 (dollars)

29. If 50% of a number is 26, what is 150% of the number?

Answer: 78

30. How many eight-pound weights are needed to balance

12 twelve-pound weights?

Answer: 18 (weights)

31. What common fraction of a class is boys if 15 of the 40 students are girls?

Answer: 5

8

32. The ratio of the number of boys to the number of girls in Ms. Moore’s class is 3:4. There are 56 students in the class. How many students are girls?

Answer: 32 (students)

33. What is the number of square meters in the area of a square if the length of a diagonal is 18 meters?

18 m

Answer: 162 (square meters)

34. Walking at a constant pace, Jill walks 5 miles in 1 hour and 20 minutes. At this rate, how many minutes does it take her to walk 1 mile?

Answer: 16 (minutes)

35. What is the number of square centimeters in the area of a square whose perimeter is 12 cm?

Answer: 9 (square centimeters)

36. Football tickets cost $13.50 each. What is the maximum number of tickets Jane can buy with $100.00?

Answer: 7 (tickets)

37. What is the value of 100 – 81 + 64 – 49 + 36 – 25 + 16 – 9 + 4 – 1?

Answer: 55

38. The sum of two positive integers is 124. What is the value of the greatest possible quotient of these two integers?

Answer: 123

39. What is the probability of rolling a sum of 7 or 11 by tossing two fair dice? Express your answer as a common fraction.

Answer: 2

9

40. What is the value of 3 • 3 + 3(3+3) – 33?

Answer: 0

41. If 9 is added to three times a number, the result is 90. What is five times the number divided by 9?

Answer: 15

42. Joe’s bowling scores were 112, 182 and 142. What does he need to score in his fourth game to get a mean score of 132?

Answer: 92

43. After losing 10% of his weight, a wrestler weighed 135 pounds. What was his original weight in pounds?

Answer: 150 (pounds)

44. Twice a number is one-halfof 48. What is the number?

Answer: 12

45. A map is drawn to a scale of 3/4 in = 100 mi. How many inches are in the line segment used to represent 800 miles?

Answer: 6 (inches)

46. In rectangle ABCD with diagonals AC and BD intersecting at E, what is the ratio of the area of BEC to the area of BDC? Express your answer as a common fraction.

A B

CD

E

Answer: 1

2

47. What is the range of the set {112, 124, 136, 148, 160} ?

Answer: 48

48. The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers?

Answer: 600

49. In how many ways can a person enter a room with 5 doorways and leave through a different doorway?

Answer: 20 (ways)

50. If , what is the

arithmetic mean of

and ?

m 0mm3

mm– 3

Answer: 1

51. A worker bee collects

enough nectar in its lifetime to

make of a pound of honey.

How many bees are needed to

collect 5 pounds of honey?

110

12

Answer: 55 (bees)

52. A box is one-third full of marbles. If one-fourth of the marbles are removed, what percent of the box is filled with marbles?

Answer: 25 (percent)

53. A pentagonal train is created by adjoining congruent pentagons with sides of 1 unit each. A pentagonal train of 10 pentagons is constructed. What is the number of units in the perimeter of the resulting polygon?

Answer: 32 (units)

54. What is the base-ten number represented by the base-eight number 31?

Answer: 25

55. Paul plans to rotate the five tires on his new four-wheel truckso that each tire will have the same number of miles when the odometer reads 20,000 miles. How many miles will be on each of the tires?

Answer: 16,000 (miles)

56. The value of the determinant

What is the value of x if

a b

c dad bc.

2 3x 6

Answer: –8

57. It takes Bryan one hour to dig a hole that is 1.4 meters wide, 1.5 meters long and 0.5 meters deep. At the same rate, how many hours will it take Bryan to dig a hole that is 2.8 meters wide, 3.0 meters long and 1.0 meters deep?

Answer: 8 (hours)

58. The city of Alexandria posted a high temperature of 18o F and a low temperature of –5o F on the same day. By how many degrees did the high temperature exceed the low temperature?

Answer: 23 (degrees)

59. Given a b = a2 + 2ab + b2, what is the value of a b when a = 4 and b = 6?

Answer: 100

60. A car travels 30 miles per gallon of gasoline. If gasoline costs $1.25 per gallon, how many dollars will it cost to travel 480 miles?

Answer: 20 (dollars)

61. From a group of four boys and two girls, two children are selected at random. What is the probability that the second child selected is a girl? Express your answer as a common fraction.

Answer: 1

3

62. The radius of a circle is 6 meters. A square is inscribed in the circle. What is the number of square meters in the area of the square?

6 m

Answer: 72 (square meters)

63. Chan’s bank account contains $450, which is 125% more than it contained at this time last year. How many dollars were in Chan’s bank account last year?

Answer: 200 (dollars)

64. A number x is 5 less than number y. Three times x is 10 less than twice y. What is the value of x – y?

Answer: – 5

65. What is the value of 113 – 3 • 112 + 3 • 11 – 1 ?

Answer: 1000

66. As shown, line k with point P lies in the same plane as circle C. How many circles in the same plane are tangent to k at P and are also tangent to circle C?

C

kP

Answer: 2 (circles)

67. Suppose that bc = 121.What is bc/2 ?

Answer: 11

68. A rectangle has a perimeter of 48 feet. What is the greatest possible number of square feet in the area of the rectangle?

Answer: 144 (square feet)

69. What is the value

of 5252 – 4752 ?

Answer: 50,000

70. How many different integers can be expressed as the sum of two different numbers in the set {1, 2, 3, 4, 5, 6}?

Answer: 9 (integers)

71. Levi had 10,000 points to exchange for prizes. Each prize he selected required half the number of points he had at the time he selected it. How many points did he have left after choosing four prizes?

Answer: 625 (points)

72. What is the remainder when 1 + 2 + 3 + 4 + ... + 9 + 10 is divided by 3?

Answer: 1

73. If a child must choose one of four entrees, one of four side dishes, and one of four drinks from a menu, how many different meals are possible?

Answer: 64 (meals)

74. The price of a radio was discounted 20% on the first day of a sale. The sale price was then discounted an additional 20% on the second day. What percent is the combined discount?

Answer: 36 (percent)

75. What is the number of square inches in the area of a square with side length 12 inches?

Answer: 144 (square inches)

76. Express in simplest form:

3 5 15 .

Answer: 15

77. What day of the week will it be 40 days from Tuesday?

Answer: Sunday

78. The volume of a cube is 343 cubic units. What is the number of square units in the surface area?

Answer: 294 (square units)

79. What is the least common multiple of 14 and 21?

Answer: 42

80. What is the sum of all positive integers t such that 1.2t < 9.6 ?

Answer: 36