2013 Chapter Competition Countdown Round

Please note that videotaping, photographing, reproducing or publishing the following questions or answers is strictly prohibited. A sample question follows that you are allowed to reproduce.

Sample Question A 3-ounce can of tomato sauce costs $1.68. In cents, what is the price per ounce?

Sample Question Answer: 56 (cents)

2013 Chapter Competition Countdown Round

1. How many CDs can Edward buy for $14 each and spend the same amount he would spend to buy 60 CDs for $7 each?

Answer: 30 (CDs)

2. What is the smallest prime that is a factor of the sum 5! + 1?

Answer: 11

3. What is the value of x if 3x + 7 = 22?

Answer: 5

4. The sum of the first n odd positive integers is 64 4. The sum of the first n odd positive integers is 64. What is the value of n?

Answer: 8

5. A bookstore employee is given a 20% discount off the retail price of any book. Assuming there is no tax, how many dollars does she pay for a book with a retail price of $55?

Answer: 44 (dollars)

6. If an isosceles triangle has base angles that are each twice the measure of the smaller angle, what is the degree measure of one of the base angles?

Answer: 72 (degrees)

7. Tim’s math homework is on five consecutive pages in the math textbook, and the sum of those page numbers is 630. What is the page number of the next page after these five homework pages?

Answer: (page) 129

8. The intersection of a circular region of radius 3 inches and a circular region of radius 2 inches has area π in2. In square inches, what is the area of the total region covered by the two circular regions? Express your answer in terms of π.

Answer: 12π (in2)

9. What is the value of the product 47 × 53?

Answer: 2491

10. What is the ratio of the number of degrees in the interior angle of a regular pentagon to the number of degrees in the interior angle of a regular hexagon? Express your answer as a common fraction.

Answer:

11. Marielle buys an equal number of 50¢ and 75¢ candy bars and spends $10, not including tax. How many candy bars did she buy altogether?

Answer: 16 (candy bars)

12. A restaurant automatically adds an 18% tip to the bill 12. A restaurant automatically adds an 18% tip to the bill. If the tip was $9, what was the bill before the tip was added, in dollars?

Answer: 50 (dollars)

13. The integer 12,345 can be expressed as the sum of two prime numbers in exactly one way. What is the larger of the two primes in this sum?

Answer: 12,343

14. Let x be an integer that satisfies x4 + 24 = 105 14. Let x be an integer that satisfies x4 + 24 = 105. What is the value of x2 − 24?

Answer: −15

15. Triangle ABC has sides of length 30, 40 and 50 units 15. Triangle ABC has sides of length 30, 40 and 50 units. What is the mean, in degrees, of the measures of the three angles?

Answer: 60 (degrees)

16. The number 18 can be written as the sum of nine consecutive integers. What is the product of these integers?

Answer: 0

17. When expressed as a common fraction, what is the value of ?

Answer:

18. What is the probability that a randomly selected two-digit positive integer will be a multiple of 11? Express your answer as a common fraction.

Answer:

19. What is the greatest prime factor of the difference 642 − 612?

Answer: 5

20. The three circles in this figure are all tangent 20. The three circles in this figure are all tangent. Their centers are collinear. The diameter of the smallest circle is that of the largest circle. What fraction of the largest circle is gray? Express your answer as a common fraction.

Answer:

21. What is the greatest possible value of ab where a, b and c are distinct positive integers less than 4? c

Answer: 9

22. If the five points marked by dots on the number line below are equally spaced, what is the value of y? y x 3 2x

Answer: 6

23. What is the number of square meters in the area of a square if the length of a diagonal is 14 meters?

Answer: 98 (m2)

24. How many miles are traveled driving at 30 mi/h for 30 minutes?

Answer: 15 (miles)

25. What is the median of the set of all positive factors of 100?

Answer: 10

26. What is the value of the quotient ?

Answer: 3

27. If a2 − b2 = 10 and a − b = 2, what is the value of a + b?

Answer: 5

28. When the positive integer x is divided by each of 4, 5 and 6, it has a remainder of 3. What is the sum of the three smallest possible values of x?

Answer: 189

29. The sum of the interior angles of a convex polygon is 900° 29. The sum of the interior angles of a convex polygon is 900°. How many sides does the polygon have?

Answer: 7 (sides)

30. How many positive integers between 100 and 200 are divisible by 14?

Answer: 7 (integers)

31. What is the least value of n for which a regular n-gon has at least 17 diagonals?

Answer: 8

32. What is the value of the difference 20182 − 20132?

Answer: 20,155

33. What is the denominator when is written as a common fraction?

Answer: 20

34. What is the range of the set {121, 142, 163, 184, 106}?

Answer: 78

35. What is the value of the quotient ?

Answer: 101

36. John is 3 years older than Peter 36. John is 3 years older than Peter. Ten years ago the sum of their ages was 59. In years, how old is John now?

Answer: 41 (years old)

37. The legs of a right triangle measure inches and inches 37. The legs of a right triangle measure inches and inches. In inches, what is the length of the hypotenuse?

Answer: 1 (inch)

38. What is the value of ?

Answer: 1

39. Together three puppies and three adult dogs weigh 60 pounds 39. Together three puppies and three adult dogs weigh 60 pounds. Two adult dogs and four puppies have a combined weight of 50 pounds. If each of the puppies has the same weight and each of the adult dogs has the same weight, how many pounds does one puppy weigh?

Answer: 5 (pounds)

40. How many distinct factors of 25! are prime numbers?

Answer: 9 (factors)

41. A regular hexagon of side length 4 inches is divided into four triangles by three nonintersecting diagonals. What is the average area, in square inches, of each of the four triangles? Express your answer in simplest radical form.

Answer: 6 (in2)

42. What is the value of x if x = ? Express your answer as a common fraction.

Answer:

43. Sharon drives 30 mi at 60 mi/h and 60 mi at 30 mi/h 43. Sharon drives 30 mi at 60 mi/h and 60 mi at 30 mi/h. What is Sharon’s average speed, in miles per hour, for the whole trip?

Answer: 36 (mi/h)

44. The three angle measures of a triangle are , and degrees 44. The three angle measures of a triangle are , and degrees. What is the degree measure of the smallest angle?

Answer: 30 (degrees)

45. What is the median of the set consisting of the first ten prime numbers?

Answer: 12

46. The sum of the squares of four consecutive positive integers is 366. What is the largest of the consecutive integers?

Answer: 11

47. The perimeter of a rhombus is 156 cm 47. The perimeter of a rhombus is 156 cm. The shorter diagonal measures 30 cm. What is the length, in centimeters, of the longer diagonal?

Answer: 72 (cm)

48. What is the smallest positive integer a such that is an integer?

Answer: 6

49. In the game Yahtzee®, five standard dice are tossed 49. In the game Yahtzee®, five standard dice are tossed. What is the probability that all five dice show even numbers or all five show odd numbers? Express your answer as a common fraction.

Answer:

50. What is the value of 0.1 + 0.2 + 0.3 + 0.4 ? Express your answer as a common fraction.

Answer:

51. Five couples went to the movies together 51. Five couples went to the movies together. They all sit in ten adjacent seats in the same row. How many different ways can they be seated if each couple sits together?

Answer: 3840 (ways)

52. What is the value of 8 ÷ 8 − 8 + 8 × 8?

Answer: 57

53. What is the product of the least common multiple and the greatest common factor of 84 and 105?

Answer: 8820

54. Kevin has twice as many cookies as Aidan and half as many cookies as Beth. If Aidan and Beth have 35 cookies together, how many cookies does Kevin have?

Answer: 14 (cookies)

55. A round birthday cake was divided into 16 congruent slices, and of the cake was eaten. The next day, five slices were eaten. How much of the original cake is left? Express your answer as a common fraction.

Answer:

56. What is the least of three consecutive integers whose product is 1716?

Answer: 11

57. Each of the 2898 students at Descartes Middle School voted for his or her favorite meal chosen from a list of six different meals. Pia created a circle graph to represent the data. If a 60-degree sector of the circle graph represents votes for pizza, how many students voted for pizza?

Answer: 483 (students)

58. The cost of daily school lunch increased from $1. 80 to $2. 25 58. The cost of daily school lunch increased from $1.80 to $2.25. What was the percent increase?

Answer: 25 (percent)

59. When expressed as an integer, what is the units digit of 2013!?

Answer: 0

60. Sun-Li sold candy bars for the glee club for $0. 50 each 60. Sun-Li sold candy bars for the glee club for $0.50 each. If she sold of the candy bars that she had and has 12 candy bars left, how many dollars has she collected so far?

Answer: 18 (dollars)

61. For what value of n is the sum of the first n positive integers equal to 190?

Answer: 19

62. An isosceles triangle with sides of integer length has a perimeter of 24 inches. If the ratio of two of its sides is 2:3, what is the number of inches in the length of one of the legs?

Answer: 9 (inches)

63. If g = 4, what is the value of ?

Answer: 5

64. A conical pool takes 2 hours to be filled at a uniform rate to a depth of 6 ft. How many minutes does it take to fill it to a depth of 3 ft?

Answer: 15 (minutes)

65. For how many integers x is (x − 3)(x + 4) < 0?

Answer: 6 (integers)

66. How many diagonals can be drawn in a convex dodecagon?

Answer: 54 (diagonals)

67. Eduardo is writing math problems 67. Eduardo is writing math problems. He writes 1 problem on day 1, 2 problems on day 2, 3 problems on day 3, and so on. If this pattern continues, on what day does he write the 50th problem?

Answer: (day) 10

68. What is the 100th term of the arithmetic sequence 3, 11, 19, 27, …?

Answer: 795

69. Richard is thinking of three distinct, positive integers 69. Richard is thinking of three distinct, positive integers. He tells Barbara their sum is 9, and he tells Lori that their product is 24. What is the median of Richard’s three numbers?

Answer: 3

70. Two sides of a triangle measure 9 units and 11 units 70. Two sides of a triangle measure 9 units and 11 units. In units, what is the positive difference between the measures of the smallest and the largest possible integral lengths of the third side of the triangle?

Answer: 16 (units)

71. What is the sum of all positive integers from 1 to 30, inclusive, that are neither multiples of 2 nor perfect squares?

Answer: 190

72. In miles, how far will Jill travel if she drives for 3 hours at an average speed of 44 mi/h?

Answer: 154 (miles)

73. What is the mean of the first five triangular numbers?

Answer: 7

74. Molly has eight U. S. coins with a total value of 78 cents 74. Molly has eight U.S. coins with a total value of 78 cents. She does not have any half-dollars. How many dimes does Molly have?

Answer: 2 (dimes)

75. The domain of a function f(x) is all real numbers and the range of f(x) is all real numbers from −1 to 12, inclusive. What is the maximum value of g(x) if g(x) = 2f(x – 1) + 3?

Answer: 27

76. In an arithmetic progression the first term is 0 and the fifth term is 5. What is the third term? Express your answer as a common fraction.

Answer:

77. The positive real numbers w, x and y satisfy the equation = 16yw. If y is tripled and w is halved, by what percent must x be increased so that the new values of w, x and y also satisfy the equation?

Answer: 50 (percent)

78. What is the value of 6. 9 × 1015 ÷ (2. 3 × 107) × 7. 0 × 10−4 78. What is the value of 6.9 × 1015 ÷ (2.3 × 107) × 7.0 × 10−4? Express your answer in scientific notation.

Answer: 2.1 × 105

79. How many sides does a regular polygon have if the measure of an interior angle is 150 degrees?

Answer: 12 (sides)

80. If g(x) = 2x – 8 and f(x) = 3x2 + 17x, what is f(g(3))?

Answer: −22