1. MATHCOUNTS®
2000 State Competition
Countdown Round

3. How many different sets of three numbers, each number chosen from the integers 1-16, could be the side lengths of a right triangle?

4. Answer: 4 (sets)

5. BC is parallel to the segment through A, and AB = BC
BC is parallel to the segment through A, and AB = BC. What is the number of degrees represented by x? A x°
124° B C

6. Answer: 28 (degrees)

7. What is the difference between 32% of 24 and 24% of 32?

8. Answer: 0

9. 4 5
A ball rebounds to of its previous height. What is the number of feet in the height of the third bounce if the ball was dropped from a height of 5 feet? Express your answer as a decimal to the nearest tenth.

10. Answer: 2.6 (feet)

11. A square is drawn inside a circle so that one vertex lies at the center of the circle and another vertex lies on the circumference. The area of the circle is 72 square inches. How many square inches are in the area of the square?

12. Answer: 36 (square inches)

13. A positive number is entered on a calculator
A positive number is entered on a calculator. Jamie pressed the x2 key and got a result of What answer would have been displayed had Jamie pressed the square root key? Express your answer as a decimal to the nearest tenth.

14. Answer: 0.5

15. What is the least number greater than 9000 that is divisible by 11?

16. Answer: 9009

17. Joel is three times as old as he was 6 years ago
Joel is three times as old as he was 6 years ago. In how many years will he be twice as old as he is now?

18. Answer: 9 (years)

19. Circle C has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle C?

20. Answer: 36 (square centimeters)

21. Sam drove 25 kilometers farther than Susan
Sam drove 25 kilometers farther than Susan. If Susan had driven twice as far as she did, Sam would have driven only 10 kilometers farther. How many kilometers did Sam drive?

22. Answer: 40 (kilometers)

23. The three sides of a right triangle have integer lengths
The three sides of a right triangle have integer lengths. The hypotenuse is 1 foot longer than the longer leg, and the shorter leg is 9 feet. What is the number of feet in the perimeter of the triangle?

24. Answer: 90 (feet)

25. What is the remainder when the product of the first 25 prime numbers is divided by 4?

26. Answer: 2

27. For positive numbers x and y, x2 + 2xy + y2 = 36
For positive numbers x and y, x2 + 2xy + y2 = 36. What is the mean of x and y?

28. Answer: 3

29. The circumference of a circle is increased by 10%
The circumference of a circle is increased by 10%. By what percent will the area of the circle increase?

30. Answer: 21 (percent)

31. At a picnic of at least three people, a plate of 26 cookies is passed around. Each person takes one cookie and passes the plate to the next person. Jana gets the first cookie and the next-to-last cookie. What is the median of all possible numbers of people at the picnic?

32. Answer: 7 (people)

33. Equilateral triangles are constructed on each side of a right triangle
Equilateral triangles are constructed on each side of a right triangle. The areas of the triangles constructed on the legs of the right triangle are 36 cm2 and 64 cm2. What is the number of square centimeters in the area of the triangle constructed on the hypotenuse?

34. Answer: 100 (square centimeters)

35. For how many two-digit positive integers is the tens digit twice the units digit?

36. Answer: 4 (integers)

37. Farmer Flagg has 400 feet of fencing to enclose a circular garden and a second 400 feet of fencing to enclose a square garden. What is the ratio of the area of the circular garden to the area of the square garden? Express your answer as a common fraction in terms of .

38. Answer: 4 

39. 1 8 1 4 1 2
In a game of chance, there is a chance of winning a car, a chance of winning a sofa, and a chance of winning a goat. What is the probability that a player will not win anything? Express your answer as a common fraction.

40. Answer: 1 8

41. The tens digit of a two-digit number exceeds the units digit by 1
The tens digit of a two-digit number exceeds the units digit by 1. If the number is divided by the sum of its digits, the quotient is 6. What is the number?

42. Answer: 54

43. How many solutions does the equation |x + 2| = 2x have?

44. Answer: 1 (solution)

45. The average of 15 consecutive positive integers is 12
The average of 15 consecutive positive integers is 12. What is the greatest of these integers?

46. Answer: 19

47. A six-question test has three true/false questions and three multiple choice questions with four choices each. If Ojeni guesses at all the answers, what is the probability that she has all answers correct? Express your answer as a common fraction.

48. Answer: 1
512

49. What is the smallest natural number greater than 1 + + + + + ?3 1 6 1 10 1 15 1 21

50. Answer: 2

51. For what value of x will a line which passes through the points (5,3) and (x,7) have a slope of - ?1 3

52. Answer: -7

53. Each term of a sequence is generated by adding the two previous terms
Each term of a sequence is generated by adding the two previous terms. What is the sum of the values of the four missing terms in the following sequence?
-12, __, -5, __, -3, __, -4, __, -9, -14, -23

54. Answer: 3

55. A red number cube and a white number cube, each with faces numbered 1 to 6, are rolled. What is the probability that the red cube shows a 4 or a 5 and that the sum of the numbers shown on the two cubes is greater than 9? Express your answer as a common fraction.

56. Answer: 1 12

57. For what value of x does 128x + 2 = ( ) x
For what value of x does 128x + 2 = ( ) x ? Express your answer as a common fraction. 1 64

58. Answer: - 14 13

59. Two whole numbers have a sum of 25
Two whole numbers have a sum of 25. What is the greatest possible product of the two numbers?

60. Answer: 156

61. What is the number of centimeters in the length of AB in the right rectangular prism shown?
9 cm
12 cm
20 cm B A

62. Answer: 25 (centimeters)

63. What is the probability that a randomly selected two-digit whole number is not divisible by 2 or 3? Express your answer as a common fraction.

64. Answer: 1 3

65. Find the positive difference between the solutions to | - 6| = 20.x 2

66. Answer: 80

67. For what value of m does (3!)(2!)(m) = 6! ?

68. Answer: 60

69. A sphere is cut into four congruent wedges
A sphere is cut into four congruent wedges. The circumference of the sphere is 12 inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of .

70. Answer: 72 (cubic inches)

71. What is the number of square units in the triangular region bounded by the lines y = 2x, y = -2x, and x = 5?

72. Answer: 50 (square units)

73. Given that f (x) = x2 - 2x + 5 and g(x) = x + 3, what is the value of f (g(5)) - g( f (5))?

74. Answer: 30

75. Two binary operations are defined by the rules a  b = a3 - b3 and a  b = (a + b)3. What is the value of (2  3)  9?

76. Answer: -1000

77. Mae ran one mile in 10. 5 minutes
Mae ran one mile in minutes. How many minutes per mile must she average for the next four miles to run five miles in 47.5 minutes? Express your answer as a decimal to the nearest hundredth.

78. Answer: 9.25 (minutes per mile)

79. Six bippitties equal three bops, and six bops equal two boos
Six bippitties equal three bops, and six bops equal two boos. How many boos are there in one bippitty, one bop and one boo? Express your answer as a common fraction.

80. Answer: 3 2

81. Twelve colored balls are placed in a bag
Twelve colored balls are placed in a bag. Six of the balls are red, four are white, and two are green. If two balls are drawn from the bag without replacement, what is the probability that at least one of them is white? Express your answer as a common fraction.

82. Answer: 19 33

83. Choose a 3  3 array of dates from the February calendar shown
Choose a 3  3 array of dates from the February calendar shown. In the array, what is the positive difference between the product of the dates in the upper-right and lower-left corners and the product of the dates in the upper-left and lower-right corners?
FEBRUARY

84. Answer: 28

85. Suli will walk to school along the paths shown
Suli will walk to school along the paths shown. The numbers on the grid represent the time in minutes required to travel along each path. What is the least number of minutes it will take Suli to get to school? 6 7 2 5 4 3
School
Home

86. Answer: 14 (minutes)

87. How many CDs which cost $7
How many CDs which cost $7.98 each plus 4% sales tax can Kylie buy with $84?

88. Answer: 10 (CDs)

89. An isosceles trapezoid has the dimensions shown
An isosceles trapezoid has the dimensions shown. How many square centimeters are in the area of the trapezoid?
50 cm
39 cm
80 cm

90. Answer: 2340 (square centimeters)

91. There are 7 boys and 9 girls in the school choir
There are 7 boys and 9 girls in the school choir. Two of the students are chosen at random to sing a duet. What is the probability that both students chosen are girls? Express your answer as a common fraction.

92. Answer: 3 10

93. Anna and Bjorn each choose an integer 1-6
Anna and Bjorn each choose an integer 1-6. What is the probability that Anna’s number is less than Bjorn’s number? Express your answer as a common fraction.

94. Answer: 5 12

95. A bag contains three red marbles, two blue marbles, and one white marble. Two marbles are randomly drawn without replacement. What is the probability of drawing one red and one blue marble? Express your answer as a common fraction.

96. Answer: 2 5

97. The lengths of the legs of a right triangle are 12 meters and 16 meters. What is the number of meters in the altitude to the hypotenuse? Express your answer as a common fraction.

98. Answer: 48 5

99. John jogged from his house to the gym at 6 miles per hour and walked back along the same route at 4 miles per hour. If the entire trip took him one hour, how many miles is his house from the gym? Express your answer as a decimal to the nearest tenth.

100. Answer: 2.4 (miles)

101. The painted area of a rectangular canvas is 72 in2, and there is a 4-inch unpainted border on all sides. The dimensions of the canvas are whole numbers. What is the number of inches in the least possible perimeter of the border?

102. Answer: 66 (inches)

103. The rectangle is inscribed in the circle, and the rhombus is inscribed in the rectangle. If the radius of the circle is 4 ft, how many feet are in the perimeter of the rhombus?

104. Answer: 16 (feet)

105. A nine-digit number uses the digits 1-9 once each
A nine-digit number uses the digits 1-9 once each. What is the probability that the number is prime?

106. Answer: 0

107. 1 2 3 4
A bin in the shape of a rectangular prism is 18 feet long, feet wide and 3 feet deep. There are 1 cubic feet in a bushel. How many bushels are in the bin? 1 4

108. Answer: 837 (bushels)

109. Jim has the same number of nickels, dimes and quarters in his pocket
Jim has the same number of nickels, dimes and quarters in his pocket. If he has $4.80, how many dimes does he have?

110. Answer: 12 (dimes)

111. For each 10 bottles recycled, 1 new bottle can be produced
For each 10 bottles recycled, 1 new bottle can be produced. Over time, how many new bottles will be produced by recycling 1000 bottles, if all recycled bottles are again recycled?

112. Answer: 111 (bottles)

113. If a = b - 1, then what is the value of |a - b| + |b - a|?

114. Answer: 2

115. One watch loses 2 minutes every hour, and the other gains 2 minutes every hour. If they start at the same time, in how many hours will the fast watch be one hour ahead of the slow watch?

116. Answer: 15 (hours)

117. If x is 25% of y, and y is 80% of z, what percent of z is x?

118. Answer: 20 (percent)

119. A circle with a radius of 5 centimeters is inscribed in an equilateral triangle. What is the number of centimeters in the altitude of the triangle?

120. Answer: 15 (centimeters)

121. What is the arithmetic mean of the consecutive natural numbers from 1 to 201 inclusive?

122. Answer: 101

123. Stetson runs one lap on a circular track which is 10 meters wide
Stetson runs one lap on a circular track which is 10 meters wide. What is the difference in the number of meters traveled if he runs on the outside of the track rather than on the inside of the track? Express your answer in terms of .

124. Answer: 20 (meters)

125. The perimeter of a rectangle with integer side lengths is 14 units
The perimeter of a rectangle with integer side lengths is 14 units. What is the positive difference between the number of square units in the greatest and least possible areas for the rectangle?

126. Answer: 6 (square units)

127. What is the fifth term of the geometric sequence whose first term is and whose third term is ? Express your answer as a common fraction. 1 4 1
100

128. Answer: 1
2500

129. A triangle with integer side lengths has two sides that measure 7 cm and 12 cm. What is the number of centimeters in the greatest possible perimeter for the triangle?

130. Answer: 37 (centimeters)

131. The average of five numbers is 42
The average of five numbers is 42. The first number is increased by 1, the second number is increased by 2, and so on, until the fifth number is increased by 5. What is the average of the new numbers?

132. Answer: 45

133. 1 3
Three boys, all working at the same rate, can mow of the town soccer fields in 8 hours. How many hours will it take four boys, mowing at the same rate, to mow the rest of the fields?

134. Answer: 12 (hours)

135. A rectangular prism whose dimensions are whole numbers has a volume of 300 cubic units. What is the smallest possible whole number that could not be one of the dimensions?

136. Answer: 7

137. Jennifer is thinking of an integer from 1 to 17 inclusive
Jennifer is thinking of an integer from 1 to 17 inclusive. Rob will guess an integer, and Jennifer will tell him if his guess is higher or lower than her number. What is the least number of guesses Rob must make to guarantee that he will know Jennifer's number?

138. Answer: 4 (guesses)

139. In which quadrant is the intersection of the lines 2x + 3y = 15 and 3x - 2y = -6?

140. Answer:(Quadrant) I (or first)

141. Solve for n: 33n +1 = 1
243

142. Answer: -2

143. What is the probability that a randomly-chosen positive integer less than 50 is a perfect cube or a perfect square?

144. Answer: 9 49

145. What is the probability that a randomly-chosen three-digit number contains the digit 4 exactly twice? Express your answer as a common fraction.

146. Answer: 13
450

147. If 25x = 36, what is the value of 5x?

148. Answer: 6

149. What is the sum of all the solutions of (x + 8)2 = ( )-4 ?1 2

150. Answer: -16

151. The median to the hypotenuse of a right triangle is 6. 5 cm long
The median to the hypotenuse of a right triangle is 6.5 cm long. What is the number of centimeters in the length of the hypotenuse?

152. Answer: 13 (centimeters)