1. 1990-1991 State Countdown Round
MATHCOUNTS
State Countdown Round

2. How many of the even counting numbers less than 100 are prime?

3. 1

4. The probability of rain on any given day in Atlanta is 20%
The probability of rain on any given day in Atlanta is 20%. After how many days would you expect it to have rained on 30 days?

5. 150 (days)

6. Two fair tetrahedral (four-sided) dice are thrown
Two fair tetrahedral (four-sided) dice are thrown. If the faces of each are numbered 3, 5, 7, and 9, what is the probability that the sum of the numbers on the bottom faces is a prime?

8. What is the shortest distance that can be traveled if you start at any point, A, B, C, or D and visit the other three points once? A D 3 B C

9. 13

10. How many positive integer pairs (x, y) satisfy A = x + y, 3 when A = 2?

11. 5 (pairs)

12. If the digit 7 is placed to the right of a three-digit number, forming a four-digit number, the value of the original number increases by 7,000. What was the original number?

13. 777

14. Fifty cards numbered 1 to 50 are placed in a box
Fifty cards numbered 1 to 50 are placed in a box. If a card is selected at random, what is the probability that the card is a prime number and a multiple of seven? Express your answer as a common fraction.

15. 1 50

16. What is the last digit in the product of all natural numbers between 1 and 100?

18. If 5! is expressed as the product of its prime factors in exponential form, what is the sum of its exponents?

19. 5

20. This table shows the distribution of scores of Mr
This table shows the distribution of scores of Mr. Sampson’s 10-point quiz. What is the median score?
Test Scores
Frequency 10 I 9 II 8
III 7
IIII 6 5 4 3 2

21. 7

22. A certain type of 8 ½“ by 11” paper has a thickness of 0. 02 cm
A certain type of 8 ½“ by 11” paper has a thickness of 0.02 cm. If 15,000 sheets of paper are stacked on top of each other, what is the height, in meters, of the stack?

23. 3 (meters)

24. Points A, B, C, and D are located on AB such that AB = 3AD = 6BC
Points A, B, C, and D are located on AB such that AB = 3AD = 6BC. If a point is selected at random, what is the probability that it is between C and D? Express your answer as a common fraction. A D C B

25. 1 2

26. Three identical squares are placed side by side to form a rectangle with a perimeter of 104 inches. What is the area, in square inches, of each square?

27. 169 (sq. in.)

28. Brad is less than 30 years old. His age is a multiple of 5
Brad is less than 30 years old. His age is a multiple of 5. Next year his age will be a multiple of 7. How old is Brad now?

29. 20 (years old)

30. How many different triangles are in the figure?

31. 13 (triangles)

32. How many even numbers are between 202 and 405?

33. 101

34. What is the positive square root of 16x16?

35. 4x8

36. When n is divided by 5, the remainder is 1
When n is divided by 5, the remainder is 1. What is the remainder when 3n is divided by 5?

37. 3

38. Express 10% of 30% of 50 as a decimal.

39. 1.5

40. A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, …}. What fractional part of the numbers in set A are not divisible by 3?

41. 2 3

42. Six softball teams are to play each other once
Six softball teams are to play each other once. How many games are needed?

43. 15 (games)

44. Two ropes, 18 meters in length and 24 meters in length, need to be cut into pieces which are all the same length. What is the greatest possible length of each piece?

45. 6 (meters)

46. What value of n makes this sentence true? n = n + 1 6 9

47. 2

48. If f(x) = 3x + 2 and g(x) = (x – 1)2, what is f(g(-2))?

49. 29

50. If March 1 is a Monday, what day of the week will it be 270 days later?

51. Friday

52. Given x = 3 and y = 2, simplify: 2x3 – 3y2 6

53. 7

54. Simplify: 7! !3!

55. 35

56. In how many different orders can a group of six people be seated around a round table?

57. 120 (ways)

58. How many different arrangements are there using the letters in the word PARALLEL?

59. 3,360 (arrangements)

60. What is √200 to the nearest tenth?

61. 14.1

62. What is the positive geometric mean of 8 and 18?

63. 12

64. Find the midpoint of the line segment with endpoints (5, -5) and ( -5, 5).

65. (0, 0)

66. Give the decimal expression for q% of q.

67. 0.01q2

68. Solve for x: 2x + 2 = 128

69. 5

70. The area of a triangle is 600 square feet
The area of a triangle is 600 square feet. Find the altitude, in feet, of the triangle if the length of the base is 30 feet.

71. 40 (meters)

72. The longest side of a right triangle is 5 feet and the shortest side is 3 feet. What is the area of the triangle in square meters?

73. 6 (sq. meters)

74. If each side of a square is increased by 50%, what is the percent of increase in the area of the square?

75. 125%

76. Four lines are drawn in a plane
Four lines are drawn in a plane. What is the maximum number of distinct regions in which the lines could divide the plane?

77. 11

78. Simplify: .6 + .3 .9 (all numbers are repeating decimals)

79. 1

80. What percent of 80% of a number will lave the number unchanged?

81. 125%

82. Find the number of square units in the area of the shaded region (0, 30) (20, 30) (30, 30) (30, 20) (0, 10) (0, 0) (10, 0) (30, 0)

83. 500 (sq. units)

84. If y = -7 + √(5 – x), find x when y = 3

85. - 95

86. In quadrilateral ABCD, angle BAD and angle CDA are trisected as shown
In quadrilateral ABCD, angle BAD and angle CDA are trisected as shown. What is the degree measure of angle AFD? B C F x E y A x X y y D

87. 80 (degrees)

88. Find the number of square units in the area of the triangle. (-2, 6)
Find the number of square units in the area of the triangle (-2, 6) (-6, 2) x (0, 0) y

89. 32 (sq. units)

90. What is the smallest integer that satisfies |2x – 3|< 8?

91. - 2

92. How many two-digit whole numbers are divisible by 14?

93. 7

94. What percent of the whole numbers less than 100 have no remainders when divided by 5?

95. 20 (%)

96. What is the 100th odd whole number?

97. 199

98. ABCD is a square 4 inches on a side, and each of the inside squares is formed by joining the midpoints of the outer square’s sides. What is the area of the shaded region?

99. 4 (sq. inches)

100. If all multiples of 3 and all multiples of 4 are removed from the list of numbers 1 through 100, how many whole numbers are left?

101. 50

102. Divide 4abc by 2a2b d2

103. 6cd2 a

104. What is the simplest radical form of 3√4  6√2

105. √2

106. Two fair cubical dice are tossed
Two fair cubical dice are tossed. What is the probability, expressed as a common fraction, that the sum of the numbers showing on the dice will be four?

107. 1 12

108. If a*b is defined as a + b, 2 what is the value of 6*(3*5)?

109. 5

110. If x and y are positive integers such that xy = 108, what is the least possible value of x + y?

111. 21

112. Find √√256,000

113. 40

114. Express the value of 1 of 10% of 0.001 in scientific notation. 100

115. 1 or(1.0) X 10-6

116. Find the sum of all solutions to this equation: x2 + 62 = 102

118. What is the sum of the lengths, in centimeters, of the two legs of a right triangle, if the length of the hypotenuse is 2√6 cm.?

119. √6 + 3 √2

120. How many diagonals can be drawn for a hexagon?

121. 9 (diagonals)

122. FIN!