MATHCOUNTS 2001 State Competition Countdown Round

1. A delivery service increased its fee by 10% and then decreased its fee by 10% a month later. The fee was originally $15. What was the number of dollars in the fee after the decrease? Express your answer to the nearest cent.

Answer: 14.85 (dollars)

1 4 2. If 2 inches represents 180 miles on a map, how many miles does 1 inches represent? 3 8

Answer: 110 (miles)

3. A rectangular prism, with a square base of side length 6 feet, has a volume of 162 cubic feet. What is the number of feet in the height of the prism? Express your answer as a mixed number. 6 6

Answer: 4 (feet) 1 2

4. What is the value of 3–1 + 2–2? Express your answer as a common fraction.

Answer: 7/12

5. Let’s call a prime number a reversible prime if the number formed by reversing its digits is also prime. What is the smallest reversible prime greater than 17?

Answer: 31

6. On a history exam, Alexia must answer 10 of 12 essay questions 6. On a history exam, Alexia must answer 10 of 12 essay questions. How many different sets of questions can she choose?

Answer: 66 (sets)

7. A horizontal line through point (–3, 4) and a vertical line through point (–1, –7) are drawn in a coordinate plane. What is the sum of the coordinates of the point of intersection of these two lines?

Answer: 3

8. The rectangular plot with largest possible area has a perimeter of 86 feet. The plot has whole number dimensions and is enclosed with fencing. How many square feet are in the area of the plot?

Answer: 462 (square feet)

9. A = {factors of 36}, B = {perfect squares}, and C = {odd numbers greater than 2}. What number is in all three sets?

Answer: 9

10. Three positive integers have an arithmetic mean of 26 and a median of 27. If the median is 5 less than the largest integer, what is the smallest integer?

Answer: 19

11. A container of eye drops contains 30 ml 11. A container of eye drops contains 30 ml. The drops are used three times daily, and each time 0.05 ml is used in each eye. How many days will the container last?

Answer: 100 (days)

12. What is the value of x for which |2x+3| is not positive 12. What is the value of x for which |2x+3| is not positive? Express your answer as a common fraction.

Answer: – 3 2

13. Janna’s average score on five 100-point tests was 80 13. Janna’s average score on five 100-point tests was 80. Her score on the sixth test raised her mean score two points. How many points did she score on the sixth test?

Answer: 92 (points)

14. What is the number of miles per hour in the average speed of a car that travels 210 miles in 3.5 hours?

Answer: 60 (miles per hour)

15. From St. Louis, MO, Lewis and Clark start on bicycle trips from the same point at the same time. Lewis travels east at 8 mph, and Clark travels south at 6 mph. After how many hours are they 60 miles apart?

Answer: 6 (hours)

16. Four pieces of paper, one of each of the colors red, yellow, blue and green, are randomly selected without replacement. What is the probability that the yellow piece is drawn first and the blue piece is drawn last? Express your answer as a common fraction.

Answer: 1 12

17. The diameter of a circle is increased so that the circumference increases by 20%. By what percent does the area increase?

Answer: 44 (percent)

18. For what value of x does 212 = ( )x ?

Answer: –4

19. What is the largest of four consecutive multiples of 6 if their mean is 27?

Answer: 36

20. What is the common fraction equivalent to 0.045 ?

Answer: 1 22

21. The sum of the digits of a two-digit number is 11 21. The sum of the digits of a two-digit number is 11. The difference between the number and the number with its digits reversed is 27. What is the sum of the original number and the number with its digits reversed?

Answer: 121

22. A rectangular wall is being covered with 4-inch by 4-inch square tiles. How many tiles are needed to completely cover a wall that is 5 feet by 7 feet?

Answer: 315 (tiles)

23. The product of three consecutive integers is 210. What is their sum?

Answer: 18

24. Six points lie in a plane, no three of which are collinear 24. Six points lie in a plane, no three of which are collinear. How many different triangles can be formed by connecting three of these points?

Answer: 20 (triangles)

25. How many different integers can be expressed as the sum of two different numbers in the set {9, 12, 15, 18, 21, 24, 27, 30, 33}?

Answer: 15 (integers)

26. A movie has a running time of 115 minutes 26. A movie has a running time of 115 minutes. If the movie begins at 7:25, at what time will it end?

Answer: 9:20

27. Compute: 322 – 182.

Answer: 700

28. What is the probability that a diagonal randomly selected from a regular octagon connects opposite vertices? Express your answer as a common fraction.

Answer: 1 5

29. If a + b = 18 and b + c = 12, what is the positive difference between a and c?

Answer: 6

30. The radius of a circle is decreased by 20% 30. The radius of a circle is decreased by 20%. By what percent is its area decreased?

Answer: 36 (percent)

31. A stack of 100 new one-dollar bills is 1 cm high 31. A stack of 100 new one-dollar bills is 1 cm high. What is the number of meters in the height of a stack of 1,000,000 new one-dollar bills?

Answer: 100 (meters)

32. What is the value of ? 2 3 24 125 ( . ) ¸

Answer: 128

33. The length of a face diagonal of a cube is 3 cm 33. The length of a face diagonal of a cube is 3 cm. What is the number of square centimeters in the surface area of the cube?

Answer: 27 (square centimeters)

34. An 8-foot board is cut into two pieces whose lengths are in the ratio of 7 to 5. What is the number of inches in the length of the longer piece?

Answer: 56 (inches)

35. How many integers x satisfy x2 + 4x + 4 = 0 ?

Answer: 1 (integer)

36. What is the value of ((  )  )  ? 1 2 1 2 1 2 1 2

Answer: 4

37. If Michael will have a family of four children, and male and female births are equally likely, what is the probability that Michael will have at least two girls? Express your answer as a common fraction.

Answer: 11 16

38. If 1/x + 1/y = 1/z, find the value of y when x = 25 and z = 24.

Answer: 600

39. How many different three-digit odd numbers can be formed using the digits 3, 4, 5 and 6 if digits cannot be repeated?

Answer: 12 (numbers)

40. A square piece of paper is folded in half vertically 40. A square piece of paper is folded in half vertically. The resulting figure has a perimeter of 21 inches. What is the number of square inches in the area of the original square?

Answer: 49 (square inches)

41. Bob makes $8 an hour at his job 41. Bob makes $8 an hour at his job. For any work over 40 hours during a week, he makes twice that rate. How many dollars does Bob make in a week during which he works 45 hours?

Answer: 400 (dollars)

42. How many odd positive integers are factors of 480?

Answer: 4 (integers)

43. How many inches are in the positive difference between 40 feet and 13 yards?

Answer: 12 (inches)

44. If a  b = 1 – ab and a  b = a – b, evaluate 4  (6  8).

Answer: 9

27 45. The space diagonal of a cube is cm. How many cubic centimeters are in the volume of the cube? 27

Answer: 27 (cubic centimeters)

46. Wynona bought 4 video tapes and 2 video discs 46. Wynona bought 4 video tapes and 2 video discs. The cost of the 4 tapes was the same as the cost of the 2 discs. If she spent a total of $104 and each tape was the same price, how many dollars did each tape cost?

Answer: 13 (dollars)

47. Two numbers, one of which is 3 more than twice the other, have a sum of 66. Find the larger of the two numbers.

Answer: 45

48. How many five-digit zip codes exist such that each of the first three digits is greater than 4?

Answer: 12,500 (zip codes)

49. Kathy ran up a hill at 4 miles per hour and down the same hill at 6 miles per hour. How many miles per hour was her average speed? Express your answer as a decimal to the nearest tenth.

Answer: 4.8 (miles per hour)

50. If 25  83  162 = 4m, what is the value of m?

Answer: 11

51. What is the ratio of the number of space diagonals of a cube to the number of edges? Express your answer as a common fraction.

Answer: 1 3

52. The measures of the three angles in a triangle form an arithmetic sequence. What is the number of degrees in the measure of the middle angle?

Answer: 60 (degrees)

53. The sum of three numbers is 50 53. The sum of three numbers is 50. The mean of five other numbers is 14. What is the mean of all eight numbers?

Answer: 15

54. Square an integer greater than or equal to 3, and subtract 1 54. Square an integer greater than or equal to 3, and subtract 1. What is the probability that the resulting value is a prime number?

Answer: 0

55. The mean of two positive single-digit integers is obtained by placing a decimal point between the two numbers. What is the sum of the two numbers?

Answer: 9

56. The integer 216 is expressed as the sum of six consecutive odd integers. What is the largest of these integers?

Answer: 41

57. Bjorn’s height is 160 cm plus one-sixth of his height 57. Bjorn’s height is 160 cm plus one-sixth of his height. What is the number of centimeters in Bjorn’s height?

Answer: 192 (centimeters)

58. What is the sum of the three smallest positive integers that have an odd number of positive factors?

Answer: 14

59. What is the units digit of the product of the first twenty prime numbers?

Answer: 0

60. How many cubic inches are in a box that is 3/4” high, 1 ” long and 8” deep? 2

Answer: 9 (cubic inches)

61. What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice?

Answer: 17 (times)

62. The length of a side of a triangle with an area of 36 square inches is 4.2 inches. What is the number of square inches in the area of a similar triangle whose corresponding side measures 5.6 inches?

Answer: 64 (square inches)

63. What is the value of (895 – 54)2 – (54 – 895)2?

Answer: 0

64. In the rectangular prism shown, AB = 6 cm, BC = 8 cm, and HC = 24 cm. What is the number of centimeters in the length of diagonal FC? A F E H C B

Answer: 26 (centimeters)

65. What is the value of Express your answer in scientific notation. ( ) ( ´ ´ ) 6 . 4 10 5 2 . 8 10 3 ? 65. What is the value of Express your answer in scientific notation. - ´ 2 ´ . 4 10 32 10 1

Answer: 1.4  107

66. The ratio of birds to squirrels in a tree is 2:3 66. The ratio of birds to squirrels in a tree is 2:3. If there are 16 birds, how many squirrels are there?

Answer: 24 (squirrels)

67. The chess club at Madison Middle School has 12 members 67. The chess club at Madison Middle School has 12 members. The club must select from its members a president, a vice president and a treasurer. In how many ways can this be done if no member may hold more than one office?

Answer: 1320 (ways)

1 3 68. A larger box is 25% longer, 20% wider and 33 % taller than a smaller box. The volume of the larger box is what percent of the volume of the smaller box?

Answer: 200 (percent)

69. What is the value of (262 – 242 – 10)2 – 102?

Answer: 8000

70. A real number m is randomly selected such that 0 < m < 8 70. A real number m is randomly selected such that 0 < m < 8. What is the probability that m < 2? Express your answer as a common fraction.

Answer: 1 4

2 3 71. When flipped, an unfair coin lands on heads of the time. What is the probability that two successive flips will result in one head and one tail? Express your answer as a common fraction.

Answer: 4 9

72. How many positive integers less than 1000 are both perfect squares and perfect cubes?

Answer: 3 (integers)

73. The product of two consecutive positive even integers is 288 73. The product of two consecutive positive even integers is 288. What is the greater of the two integers?

Answer: 18

74. What is the sum of the first 10 positive integers?

Answer: 55

75. What is the number of cubic meters in the volume of a cube with side length 6 m?

Answer: 216 (cubic meters)

76. What is the value of 210  5 + 105  (–9)?

Answer: 105

77. How many diagonals does a pentagon have?

Answer: 5 (diagonals)

78. Which number is not divisible by 3: 96, 133, 441 or 621?

Answer: 133

3 4 13 16 25 32 79. Which of the following numbers is greatest: , or ? Express your answer as a common fraction.

Answer: 13 16

Express your answer as a common fraction. 80. Compute: . 1 2 + Express your answer as a common fraction.

Answer: 2 3