Yavapai County Math Contest College Bowl Competition. January 28, 2010

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1 Yavapai County Math Contest College Bowl Competition January 28, 2010

2 Is your adrenalin engaged?

3 What is ? 82

4 Solve for x in: 2x + 7 = 1 3x. x=-6/5 (or x=-1.2)

5 If a fair die is rolled once, what is the probability of observing a three? 1/6

6 What is the sum of the degree measures of the interior angles of a hexagon? 720

7 This year attendance of the contest is 20% higher than last year s attendance. If this year attendance is 300, what was last year s attendance? 250

8 Find the missing term that is needed to make the trinomial below a perfect square:? 6x + 1 9x 2

9 Evaluate the expression:

10 What is the next number in the sequence 1, 3, 7, 15, 31,...? 63

11 A square inscribed in another square has as vertices the midpoints of the sides of the bigger square. If the side of the big square is 2, find the area of the green square. 2

12 If the length of a rectangle is increased by 50% and the width is decreased by 50%, then the area is increased or decreased by what percentage? decreased by 25%

13 Write in decimal form:

14 Write the equation that best describes the relationship between x and y below. x y y = 2x 1

15 We let A$B stand for A 2 B. What is 5$(2$1)? 22

16 The diagonals of the rhombus shown are 6 cm and 8 cm. What is the perimeter of the rhombus? 20 cm

17 Find two numbers whose sum is 20 so that the difference of their squares is and 11

18 Simplify 8! 6! 56

19 What is the value of log 2 (8)? 3

20 Give an example of two numbers A and B such that A > B, but A 2 < B 2. any B < A

21 Consider a cube with side 1 inch. What is the total length of all its edges? 12 inches

22 Find the missing number: ? 35

23 Express as a common fraction in reduced form. 1/9

24 sin(α) =? 5/11

25 Find the next number in the sequence 1, 3, 6, 10, 15, 21,... 28

26 One parallelogram has sides of lengths 4 cm and 5cm. One of its diagonals is perpendicular to one side. What is its area? 12 cm 2

27 Robert s lawn is a square with side 7 m long and has a concrete sidewalk 1 m wide all around it. What is the area of the concrete? 32

28 Solve for x: 1 x 2 2 x 15 = 0 1 5, 1 3

29 Solve for x: x = 2 x=64

30 Find all values of x for which x x 1 = 2x. 1 x

31 A sphere is inscribed in a cube. What is the ratio of their volumes? 6 : π or π : 6

32 Points A to F are evenly spaced on a number line. What fraction represents the coordinate of point B? 5/12

33 f (x) = 2x + 19 is an example of a function. linear

34 Simplify: (X + Y ) 2 (X Y ) 2 4XY

35 Which of the following is equal to ? A B C D E. 3 5 A

36 What is the angle x? 41

37 Find the numbers on the last row ???

38 10 is the average of what two primes? 7 and 13 or 3 and 17

39 You know that D = gj. If D is 30 when J is 5, what is J when D is 10? 5/3

40 Of the integers from 50 to 100, how many are divisible by 3? 17

41 What is the area of an equilateral triangle having sides of length 10? 25 3

42 If I walk halfway home at 2 mph, how fast must I run the rest of the way in order to average 3 mph? 6 mph

43 What are the coordinates of the point that is the reflection of (2, 4) in the line x = 1? ( 4, 4)

44 What is the smallest 3-digit prime number? 101

45 Write the base 10 number 11 in base =

46 How long is x? 21/5

47 This is a 2-by-2 Sudoku. Find the digit in the lower right corner ? 3

48 How many distinct teams of two people could be formed from a group of four students? ( ) 4 2 = 4! 2! 2! = 6

49 Evaluate the sum

50 How many distinct divisors does the number 36 have, including 1 and 36? 9

51 Two perpendicular lines meet at (0,6). The x-intercept of one is (4, 0). What is the x-intercept of the other line? (-9,0)

52 The function f satisfies the equation: f (2x) = f (x) + x. If f (2) = 3, what is f (8)? 9

53 Find the x-intercept of the line with equation 3x + 5y = 5. (5/3, 0)

54 Find the domain of the function 1 f (x) =. x + 2 ( 2, )

55 Evaluate: ( ( ) ) 1 1 (1 + 1)

56 A cube has volume 27 cm 3. What is its surface area? 54 cm 2

57 Find the missing numbers: ????=36 and??=30

58 Simplify: ( 2 ) 12 64

59 What is the least common multiple of 45 and 75? 225

60 Replace the letters A, B, and C with digits such that the following addition holds true: AC + AC + AC = BBB. A = 3, B = 1, C = 7 or A = 7, B = 2, C = 4

61 Consider three lines in the plane, as depicted. In how many distinct regions do they divide the plane? 7

62 Bob can mow the lawn in 50 minutes. Together Bob and Tom can mow the lawn in 30 minutes. How long does it take Tom to mow the lawn alone? 75 minutes

63 Calculate: 8 (7 (6 (5 (4 (3 (2 1)))))) 4

64 What is the probability of obtaining a double six by rolling a pair of fair dice once? 1 36

65 What is the smallest number satisfying x 9 5? 4

66 The diagram shows a regular pentagon with its diagonals. If FG = x and GB = y then how long is AC? x + 2y

67 A perfect cube is a number of the form n 3, with n 1 integer. (For example, 8 is a perfect cube, because 8 = 2 3.) How many perfect cubes are there less or equal to 1, 001? 10

68 How far apart, in degrees, are the hands of a clock at 4:00? 120

69 The difference in the squares of two consecutive odd numbers is 40. What is the smaller of the two numbers? 9

70 What is the next number in the following sequence? 447, 223, 111, 55, 27,... 13

71 Triangle ABC has area 600. If M and N are the midpoints of AB and AC, respectively, what is the area of the triangle AMN? 150

72 You earn $400 per week. Your boss gives you a 25% raise. However, two weeks after this you receive a 25% pay cut due to cost overruns. What is your weekly salary now? $375

73 What is the geometric mean of 9 and 16? 12

74 The equation 4x x = 0 has two solutions. What is their average? -3

75 A teacher said that her class had 100 students, 24 males and 32 females. What number base did the teacher use in this statement? 6

76 If the plane is rotated 90 clockwise about the origin, where does the point (3, 8) end up? (8, 3)

77 Consider a function that satisfies the relation f (x + y) = f (x)f (y), for all x and y. If f (1) = 2, what is f (4)? f (4) = 16; one such function is f (x) = 2 x

78 How many prime numbers are between 20 and 30? two: 23 and 29

79 If now is noon, what will a clock read in 35 hours? (Use AM and PM.) 11 PM

80 What is the missing number? ?? 12

81 What is the angle formed by the pictured diagonals of two faces of a cube?

82 What is the angle formed by the pictured diagonals of two faces of a cube? 60 ; see the equilateral triangle

83 I am writing a tune on the computer. Voice 1 sings a pattern 4 measures long, and Voice 2 sings a part that is 5 measures long. The program starts them all singing at the same time, repeats their parts over and over, and stops when both are simultaneously at the end of their songs. How many measures is the whole tune? 20 measures

Math is Cool Masters

Math is Cool Masters Sponsored by: GENIE Industries 7 th Grade November 19, 2005 Individual Contest Express all answers as reduced fractions unless stated otherwise. Leave answers in terms of π where applicable. Do not round