MATHCOUNTS ® 2000 State Competition Countdown Round

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Transcript of MATHCOUNTS ® 2000 State Competition Countdown Round

  • Slide 1
  • MATHCOUNTS 2000 State Competition Countdown Round
  • Slide 2
  • Slide 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?
  • Slide 4
  • Answer: 4 (sets)
  • Slide 5
  • BC is parallel to the segment through A, and AB = BC. What is the number of degrees represented by x? A B C x 124
  • Slide 6
  • Answer: 28 (degrees)
  • Slide 7
  • What is the difference between 32% of 24 and 24% of 32?
  • Slide 8
  • Answer: 0
  • Slide 9
  • 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. 4545
  • Slide 10
  • Answer: 2.6 (feet)
  • Slide 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?
  • Slide 12
  • Answer: 36 (square inches)
  • Slide 13
  • A positive number is entered on a calculator. Jamie pressed the x 2 key and got a result of 0.0625. What answer would have been displayed had Jamie pressed the square root key? Express your answer as a decimal to the nearest tenth.
  • Slide 14
  • Answer: 0.5
  • Slide 15
  • What is the least number greater than 9000 that is divisible by 11?
  • Slide 16
  • Answer: 9009
  • Slide 17
  • 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?
  • Slide 18
  • Answer: 9 (years)
  • Slide 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?
  • Slide 20
  • Answer: 36 (square centimeters)
  • Slide 21
  • 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?
  • Slide 22
  • Answer: 40 (kilometers)
  • Slide 23
  • 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?
  • Slide 24
  • Answer: 90 (feet)
  • Slide 25
  • What is the remainder when the product of the first 25 prime numbers is divided by 4?
  • Slide 26
  • Answer: 2
  • Slide 27
  • For positive numbers x and y, x 2 + 2xy + y 2 = 36. What is the mean of x and y?
  • Slide 28
  • Answer: 3
  • Slide 29
  • The circumference of a circle is increased by 10%. By what percent will the area of the circle increase?
  • Slide 30
  • Answer: 21 (percent)
  • Slide 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?
  • Slide 32
  • Answer: 7 (people)
  • Slide 33
  • 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 cm 2 and 64 cm 2. What is the number of square centimeters in the area of the triangle constructed on the hypotenuse?
  • Slide 34
  • Answer: 100 (square centimeters)
  • Slide 35
  • For how many two-digit positive integers is the tens digit twice the units digit?
  • Slide 36
  • Answer: 4 (integers)
  • Slide 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 .
  • Slide 38
  • Answer: 44
  • Slide 39
  • 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. 1414 1818 1212
  • Slide 40
  • Answer: 1818
  • Slide 41
  • 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?
  • Slide 42
  • Answer: 54
  • Slide 43
  • How many solutions does the equation |x + 2| = 2x have?
  • Slide 44
  • Answer: 1 (solution)
  • Slide 45
  • The average of 15 consecutive positive integers is 12. What is the greatest of these integers?
  • Slide 46
  • Answer: 19
  • Slide 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.
  • Slide 48
  • Answer: 1 512
  • Slide 49
  • What is the smallest natural number greater than 1 + + + + + ? 1616 1313 1 10 1 15 1 21
  • Slide 50
  • Answer: 2
  • Slide 51
  • For what value of x will a line which passes through the points (5,3) and (x,7) have a slope of - ? 1313
  • Slide 52
  • Answer: -7
  • Slide 53
  • 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
  • Slide 54
  • Answer: 3
  • Slide 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.
  • Slide 56
  • Answer: 1 12
  • Slide 57
  • For what value of x does 128 x + 2 = ( ) x ? Express your answer as a common fraction. 1 64
  • Slide 58
  • Answer: - 14 13
  • Slide 59
  • Two whole numbers have a sum of 25. What is the greatest possible product of the two numbers?
  • Slide 60
  • Answer: 156
  • Slide 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
  • Slide 62
  • Answer: 25 (centimeters)
  • Slide 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.
  • Slide 64
  • Answer: 1313
  • Slide 65
  • Find the positive difference between the solutions to | - 6| = 20. x2x2
  • Slide 66
  • Answer: 80
  • Slide 67
  • For what value of m does (3!)(2!)(m) = 6! ?
  • Slide 68
  • Answer: 60
  • Slide 69
  • 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 .
  • Slide 70
  • Answer: 72 (cubic inches)
  • Slide 71
  • What is the number of square units in the triangular region bounded by the lines y = 2x, y = -2x, and x = 5?
  • Slide 72
  • Answer: 50 (square units)
  • Slide 73
  • Given that f (x) = x 2 - 2x + 5 and g(x) = x + 3, what is the value of f (g(5)) - g( f (5))?
  • Slide 74
  • Answer: 30
  • Slide 75
  • Two binary operations are defined by the rules a b = a 3 - b 3 and a b = (a + b) 3. What is the value of (2 3) 9?
  • Slide 76
  • Answer: -1000
  • Slide 77
  • Mae ran one mile in 10.5 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.
  • Slide 78
  • Answer: 9.25 (minutes per mile)
  • Slide 79
  • 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.
  • Slide 80
  • Answer: 3232
  • Slide 81
  • Twelve colored balls are