1. MATHCOUNTS ® 1999 State Competition Countdown Round

2. How many of the first 100 positive integers are neither perfect squares nor perfect cubes?

3. Answer: 88 (integers)

4. A palindrome is a number that reads the same forwards and backwards. How many palindromes are between 1000 and 10,000?

5. Answer: 90 (palindromes)

6. The surface of a 4” x 5” x 6” block is painted yellow. If the block is separated into one-inch cubes, what is the number of cubes with exactly one yellow face?

7. Answer: 52 (cubes)

8. The lengths of two sides of a right triangle are meters and meters. The number of meters in the length of the third side is a whole number. How many meters are in the third side of the triangle?

9. Answer: 8 (meters)

10. A bag contains yellow and blue marbles. The probability of selecting a yellow marble from the bag is. If there are 27 blue marbles in the bag, what is the number of yellow marbles?

11. Answer: 24 (marbles)

12. A uniform piece of lumber is 12 ft long and weighs 30 lb. If a 5 ft length is removed, what is the number of pounds in the remaining piece? Express your answer as a mixed number.

13. Answer: 17 (pounds) 1212

14. If q, r and s each represent a different integer 0-9, what is the value of s? qqqq rrrr + ssss rqqqs

15. Answer: 8

16. Compute:.

17. Answer: 10,000

18. What number is 11 % of 13 239? 1919

19. Answer: 1471

20. For what whole number n is ?

21. Answer: 5

22. What is the ratio of 0.16 to 0.83? Express your answer as a common fraction.

23. Answer: 1515

24. The measure of the angles of a pentagon are in the ratio of 3:3:3:4:5. What is the number of degrees in the measure of the largest angle?

25. Answer: 150 (degrees)

26. A digital clock shows hours and minutes. How many times during a 12-hour period will the sum of the digits on this clock be 3?

27. Answer: 12 (times)

28. It takes 6 days for a team of 6 workers to carpet a house. At the same rate, what is the minimum number of additional workers needed to carpet the same house in just 5 days?

29. Answer: 2 (workers)

30. If and, what is the value of ?

31. Answer: 114

32. John has some loaves of bread in a bag. If he had 6 more loaves, he would have three times as many. How many loaves does John have in the bag?

33. Answer: 3 (loaves)

34. If a two-digit prime number is selected at random, what is the probability that the sum of its digits is 9?

35. Answer: 0

36. In right triangle ABC, CAB is a right angle. Point M is the midpoint of BC. What is the number of centimeters in the length of median AM? Express your answer as a decimal to the nearest tenth.

37. Answer: 2.5 (centimeters)

38. What is the number of degrees in the exterior angle of the largest angle of a triangle if the interior angles are in the ratio of 2:3:5?

39. Answer: 90 (degrees)

40. A 12 feet by 8 feet rectangular Persian rug is centered within a rectangular room so that a region of constant width surrounds the carpet. Given that 96 square feet of flooring is not covered by the rug, what is the number of feet in the measure of the longer dimension of the room?

41. Answer: 16 (feet)

42. What is the least positive integer that is not a factor of 7! ?

43. Answer: 11

44. A right circular cone is inscribed in a right circular cylinder. The volume of the cylinder is 72  cubic centimeters. What is the number of cubic centimeters in the space inside the cylinder but outside the cone? Express your answer in terms of .

45. Answer: 48  (cubic centimeters)

46. Compute 12 % of 240. 1212

47. Answer: 30

48. In an isosceles trapezoid, the lengths of the bases are 8 cm and 16 cm. The length of a diagonal is 13 cm. What is the number of square centimeters in the area of the trapezoid?

49. Answer: 60 (square centimeters)

50. What is the greatest two-digit whole number that can be expressed as the product of two consecutive whole numbers?

51. Answer: 90

52. What is the greatest three-digit number that is divisible by 6?

53. Answer: 996

54. If the measure of each side of a square is decreased by 10%, by what percent is the area decreased?

55. Answer: 19 (percent)

56. What is the units digit of ?

57. Answer: 0

58. All squares have the same value, and all circles have the same value. What is the value of the sum of three circles? + + + + = 21 + + + + = 19 + + = ?

59. Answer: 9

60. Six less than a number is multiplied by 6. Eight less than the same number is multiplied by 8. The two results are equal. What is the number?

61. Answer: 14

62. When Sarah turns 36 years old, her father will be 62. Currently, Sarah is of her father’s age. How many years old is Sarah now? 1313

63. Answer: 13 (years)

64. The sum of seven consecutive even numbers is 13,902. What is the smallest of the seven numbers?

65. Answer: 1980

66. Eleven coins total $1.19, but no combination of the coins will give a value of exactly $1. How many of the coins are quarters?

67. Answer: 3 (quarters)

68. Jimi is taking a bicycle trip and needs to arrive by 6 o’clock. If he travels at an average rate of 15 mph, he’ll arrive at his destination one hour early, but if he travels at an average rate of 10 mph, he’ll arrive one hour late. How many miles away is his destination?

69. Answer: 60 (miles)

70. Mark baked cookies for the neighborhood children. He gave each child 6 cookies, and he had 7 cookies remaining. He was going to give another cookie to each child, but he was 1 cookie short. How many cookies had Mark baked?

71. Answer: 55 (cookies)

72. An eight-digit number contains two 5’s, two 6’s, two 7’s and two 8’s. The 5’s are separated by one digit, the 6’s are separated by two digits, the 7’s are separated by three digits, and the 8’s are separated by four digits. What is the greatest possible value for the number?

73. Answer: 85,756,876

74. Calculate:.

75. Answer: 27

76. Calculate and express your answer as a common fraction:

77. Answer: 105 71

78. What is the least positive integer n such that ?

79. Answer: 5

80. How many different three-member committees can be chosen from a group of ten people?

81. Answer: 120 (committees)

82. A rectangle with length 8 units and width 5 units is centered at the origin of a coordinate plane, and its longer side is parallel to the x-axis. What percent of the rectangular region is to the right of the line with equation ?

83. Answer: 75 (percent)

84. What is the smallest natural number that has exactly five factors?

85. Answer: 16

86. For what value of n does ?

87. Answer: 2

88. What is the sum of the real values of x for which the following expression is undefined? Express your answer as a common fraction.

89. Answer: 15 2

90. A 3” x 5” photograph can be enlarged to a panoramic view that is 5” x 10”. What is the ratio of the area of the original photograph to the area of the enlargement? Express your answer as a common fraction.

91. Answer: 3 10

92. The sum of two numbers is 9, and their difference is 7. What is the positive difference of the squares of these two numbers?

93. Answer: 63

94. One diagonal of a rhombus measures 48 inches. Given that one side of the rhombus has length 26 inches, what is the number of square inches in the area of the rhombus?

95. Answer: 480 (square inches)

96. A centipede climbs a 40-foot tree. Each day he climbs 5 feet, and each night he slides down 2 feet. In how many days will the centipede reach the top of the tree?

97. Answer: 13 (days)

98. For how many integers n is |n - 5| < 3 ?

99. Answer: 5 (integers)

100. Bill, Frank and Grete are playing a game. Bill is three times as likely to win the game as Frank, and Frank is two times as likely to win the game as Grete. Assuming no ties, what is the probability that Grete will win the game? Express your answer as a common fraction.

101. Answer: 1919

102. What is the greatest rational number n such that ?

103. Answer: 8

104. What is the sum of the reciprocals of 1, 2, 3, 4 and 5? Express your answer as a common fraction.

105. Answer: 137 60

106. A jar contains 60 marbles, each colored either green or yellow, all the same size. There are twice as many green marbles as yellow in the jar. How many yellow marbles must be added to the jar so the probability of choosing a green marble is ? 1212

107. Answer: 20 (marbles)

108. Each of the 30 students in a physical education class plays the same number of minutes of full- court basketball in a class period. Ten students play at a time, and the class period lasts 51 minutes. What is the number of minutes each student plays?

109. Answer: 17 (minutes)

110. A space diagonal of a given cube is inches long. What is the number of square inches in the surface area of the cube?

111. Answer: 600 (square inches)

112. A monument is constructed so that the floor is a regular hexagon with sides measuring 8 feet. What is the number of square feet in the area of the floor? Express your answer in simplest radical form.

113. Answer: (square feet)

114. In China, small spheres are used to exercise a person’s hands. If the diameter of one of these spheres measures 2 inches, how many square inches are in its surface area? Express your answer in terms of .

115. Answer: 4  (square inches)

116. For what value of n does ?

117. Answer: 10

118. Sam and Janet each have a whole number of dollars, and of Sam’s money equals of Janet’s money. The sum of their money is more than $10. What is the least whole number of dollars they could have combined? 1313 1212

119. Answer: 15 (dollars)

120. What is the greatest digit that can be placed in the box so that the six-digit number shown is divisible by 4? 352,9 2

121. Answer: 9

122. When you roll three number cubes, each numbered 1 through 6, what is the probability that the tops of the three cubes will show either all even numbers or all odd numbers? Express your answer as a common fraction.

123. Answer: 1414

124. For every 3 herks, there are 4 merks. Together, there are 84 herks and merks. How many more merks than herks are there?

125. Answer: 12 (merks)

126. Given that, evaluate. Express your answer as a decimal to the nearest hundredth.

127. Answer: 0.25

128. Mark earns $8.00 per hour for the first 8 hours he works each day. If he works more than 8 hours, he gets paid 1.5 times his hourly rate for the extra hours. On Monday, Mark worked from 7:30 AM to 6:30 PM and took an hour for lunch. If his lunch hour is unpaid, how many dollars did Mark earn on Monday?

129. Answer: 88 (dollars)

130. If three values from the list below are selected at random without replacement, what is the probability that they are equal? Express your answer as a common fraction. 40% of 1 0.4 24 60 2525 1414

131. Answer: 2525

132. Points A, B, C, D and E are five points on a line segment with endpoints A and E. The points are in the order listed above from left to right. CD = AB, BC = CD, AB = AE, and AE = 12 cm. What is the number of centimeters in the length of AD? Express your answer as a decimal to the nearest tenth. 1212 1212 1212

133. Answer: 10.5 (centimeters)

134. What is the maximum possible value of x in the inequality ?

135. Answer: 8

136. Three straight lines intersect at O, and OD bisects COE. The ratio of COB to BOF is 7:2. What is the number of degrees in m COD?

137. Answer: 20 (degrees)

138. Jena starts on a trip at 8:00 AM and travels at a rate of 50 mph. Harvey starts out at 10:30 AM the same day and travels the same route at a rate of 75 mph. What is the sum of the total miles traveled by Harvey and Jena when Harvey catches up with Jena?

139. Answer: 750 (miles)

140. A right circular cylinder with radius 2 inches and height 1 inch has the same volume as a second right circular cylinder. The radius of the second cylinder is 1 inch. What is the number of inches in the second cylinder’s height?

141. Answer: 4 (inches)

142. For how many values of x does ?

143. Answer: 2 (values)

144. Using only quarters, dimes, nickels, and pennies, what is the minimum number of coins needed to get 91 cents?

145. Answer: 6 (coins)

146. The measure of an exterior angle of a triangle is 105 o. If one of the non-adjacent interior angles measures 38 o, what is the number of degrees in the other non-adjacent interior angle?

147. Answer: 67 (degrees)

148. Given that 36 feet of fencing are needed to fence the sides of a triangular region with sides in the ratio 3:4:5, how many square feet are in the area of the triangular plot?

149. Answer: 54 (square feet)

150. Larry traveled from his house to the mall at 40 mph and returned along the same route at 60 mph. What was his average speed in miles per hour for the entire trip?

151. Answer: 48 (miles per hour)

152. How many seconds does it take the second hand of a clock to rotate through an angle of 36 o ?

153. Answer: 6 (seconds)