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PPT – 1. What is the slope of the line determined by any two solutions to the equation ? Express your answer as a common fraction. PowerPoint presentation | free to download - id: 767d8d-MjBkN

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MATHCOUNTS?

2002 State Competition Countdown Round

1. What is the slope of the line determined by

any two solutions to the equation

? Express your answer as a common fraction.

Answer

2. How many different prime factors does 20!

have?

Answer 8 (factors)

3. Murphys sells notebooks at a cost of five for

2, Right-Ade sells six for 3 and Callahans

sells twelve for 5. What is the number of cents

in the cost per notebook for the least expensive

store?

Answer 40 (cents)

4. Compute

Answer

5. A rectangle has an area of 100 square units.

Its dimensions are whole units. What is the

number of units in the greatest possible

perimeter?

Answer 202 (units)

6. What is the sum of the roots of 2x2 1 3x

? Express your answer as a common fraction.

Answer

7. The circumference of a circle increases from

20 cm to 30 cm. By how many centimeters will

the radius increase?

Answer 5 (centimeters)

8. is similar to . What is

the number of centimeters in the length of

? Express your answer as a decimal to

the nearest tenth.

B

8cm

5cm

E

C

A

3cm

F

D

Answer 4.8 (centimeters)

9. Warren buys limes at a cost of three for 50.

He then sells them at a price of five for 90.

How many limes must he sell to make a total

profit of 2?

Answer 150 (limes)

10. Solve for x

Answer

11. What is the length of the diagonal of a

square with side length cm? Express

your answer in simplest form.

Answer 100 (centimeters)

12. One leg of a right triangle is 12 inches, and

the measure of the angle opposite that leg is

30. What is the number of inches in the

hypotenuse of the triangle?

Answer 24 (inches)

13. Compute 25 (1999 2000 2001 2002).

Answer 200,050

14. How many squares of area 4 square units

have all four vertices on the points in the

6 x 6 grid of points below?

5 units

5 units

Answer 16 (squares)

15. For what integer n is

Answer 6

16. If 3 of 400 is equal to 5n 2, then

what is the value of n ?

Answer 2

17. Determine the area in square units of

parallelogram ABCD with vertices A (-6, 2), B

(12, 2), C (15, 8) and D (-3, 8).

Answer 108 (square units)

18. Determine the sum of all single-digit

replacements for n such that the number

42,789,n37 is divisible by 3.

Answer 15

19. A large cube is made from unit cubes. The

large cubes faces are then painted, and the unit

cubes are separated. Twenty-seven unit

cubes have no faces painted. How many unit cubes

were used to make the large cube?

Answer 125 (unit cubes)

20. Lola had 7 bags. Each bag contained 7 mice.

Each mouse had 7 pieces of cheese. How many

pieces of cheese were in all of Lolas bags

combined?

Answer 343 (pieces)

21. How many different integers can be written as

the sum of two distinct members of the set 1,

3, 5, 7, 9, 11, 13?

Answer 11 (integers)

22. For what value of b is the set

equal to the set 4, 5, 6?

Answer 4

23. Kristin has 5 different pairs of earrings.

What is the probability that two

randomly-selected earrings will be a matching

pair? Express your answer as a common fraction.

Answer

24. The digits 2, 3, 5 and 7 are arranged

randomly to form a four-digit number. What

is the probability that the number is odd?

Express your answer as a common fraction.

Answer

25. How many miles will a race car travel in 90

seconds at a rate of 80 miles per hour?

Answer 2 (miles)

26. Compute

Answer 60

27. Three marbles are randomly selected, without

replacement, from a bag containing two red, two

blue and two green marbles. What is the

probability that one marble of each color is

selected? Express your answer as a common

fraction.

Answer

28. A snow plow clears 16 miles of roads per

hour. If the plows rate remains constant, how

many miles of roads can it clear in 4.5

hours?

Answer 72 (miles)

29. What is the largest prime factor of 2323?

Answer 101

30. What is 10 of 20 of 30 of 10,000 ?

Answer 60

31. How many factors of are perfect

squares?

Answer 12 (factors)

32. Determine the number of square units in the

area of the trapezoid whose bases are

(5 - x) units and (13 x) units and whose

height is 10 units.

Answer 90 (square units)

33. What is the least four-digit whole number

that is both a perfect square and a perfect cube?

Answer 4096

34. How many non-empty subsets containing only

prime numbers does the set 1, 2, 3, 4, 5, 6, 7

have?

Answer 15 (subsets)

35. The following data was reported re-garding

residential recycling in Tallahas-see, Florida.

How many recycled products on the list below

experienced at least a 100 increase from 1989

to 1999? 1989

1999 Aluminum 37 tons 88 tons

Tin 117 tons 184 tons Glass

801 tons 353 tons Newspaper

1664 tons 3726 tons Plastic 93

tons 266 tons

Answer 3 (products)

36. If what is the value

of x ?

Answer 6

37. When is expressed as a repeating

decimal of the form 0.ababab..., what is the

product of a and b ?

Answer 5

38. What decimal number is exactly of the

way from .826 to .827 ?

Answer .82675

39. School uniform parts are on sale. The 25

pair of slacks can be purchased at a 15

discount, and the 18 shirt can be purchased at a

20 discount. What is the total number of

dollars spent to purchase 1 pair of slacks and 1

shirt during the sale? Express your answer to

the nearest hundredth.

Answer 35.65 (dollars)

40. A drawing of a room uses a scale of 1 inch to

1.5 feet. How many square inches of area would

be required on the drawing for a rectangular

table that measures 6 feet by 2.25 feet?

Answer 6 (square inches)

41. Lon doubled a four-digit number and then

subtracted 1275. The result was 3573. What

four-digit number did Lon double?

Answer 2424

42. How many minutes are in 10 of a 24-hour day?

Answer 144 (minutes)

43. Each row of a seating arrangement seats 7 or

8 people. Forty-six people are to be

seated. How many rows seat exactly 8

people if every seat is occupied?

Answer 4 (rows)

44. If a triangle has two sides of lengths 5 and

7 units, then how many different integer lengths

can the third side be?

Answer 9 (lengths)

45. What is the least perfect square with 3

different prime factors?

Answer 900

46. For training, a cyclist rides the same

distance 3 days a week, twice the distance 2 days

a week and three times the distance 1 day a week.

What portion of the weekly total distance did

the cyclist ride the day she rode the farthest?

Express your answer as a common fraction.

Answer

47. How many of the following numbers are greater

than 3 and less than 4?

Answer 1 (number)

48. The sum of ten consecutive odd numbers is

640. What is the greatest of these ten numbers?

Answer 73

49. Two integers have a sum of and a

product of 54. What is the smaller of these two

integers?

Answer

50. A number is defined as a falling number if

each digit in the number is less than the digit

to its left. What is the least falling number

that is greater than 97,543 ?

Answer 97,610

51. The spinner below is designed so that landing

in each of the five sectors is equally likely.

What is the probability that the sum of the

numbers determined in two spins is 10? Express

your answer as a common fraction.

Answer

52. Each of six softball teams is to play each

other team once. How many games are needed?

Answer 15 (games)

53. The endpoints of a line segment are (2, 3)

and (8, 15). What is the sum of the coordinates

of the midpoint?

Answer 14

54. What is the probability that a

randomly-selected two-digit number is a multiple

of 14? Express your answer as a common fraction.

Answer

55. Nine different two-digit numbers can be

formed with the digits 1, 3 and 7. How many of

these numbers are prime?

Answer 7 (numbers)

56. A jar has 8 red, 9 blue and 10 green marbles.

What is the least number of marbles you can

remove from the jar to guarantee that you have

three of the same color?

Answer 7 (marbles)

57. Three congruent squares are placed

side-by-side to form a rectangle with a perimeter

of 104 inches. What is the number of square

inches in the area of one of the original

squares?

Answer 169 (square inches)

58. How many of the first 500 positive

integers are divisible by 3, 4 and 5?

Answer 8 (integers)

59. Sixteen times the reciprocal of the

circumference of a circle equals the diameter of

the circle. What is the number of square units

in the area of the circle?

Answer 4 (square units)

60. Hillary has eleven coins, all dimes and

nickels. In total, the coins are worth 75 cents.

How many nickels does she have?

Answer 7 (nickels)

61. What is the reciprocal of ?

Express your answer in simplest form.

Answer 45

62. How many even numbers are greater than 202

and less than 405?

Answer 101 (numbers)

63. Compute

Answer 210

64. What percent of the integers from 1 to 100,

inclusive, leave remainder 1 when

divided by 5?

Answer 20 (percent)

65. In Mathland, the area codes consist of three

digits such that the first is a digit from 2 to

9, inclusive, the second is either 0 or 1, and

the third is a digit other than 0. How many area

codes are possible?

Answer 144 (area codes)

66. The average age of a group of doctors and

lawyers is 40. The average age of the doctors is

35, and the average age of the lawyers is 50.

What is the ratio of the number of lawyers to the

number of doctors? Express your answer as a

common fraction.

Answer

67. Mr. Sanchezs students were asked to add two

positive integers. Juan subtracted by

mistake and got 2. Maria mistakenly

multiplied and got 120. What was the correct

answer?

Answer 22

68. What is the sum of all positive integer

divisors of 77?

Answer 96

69. The length of a diagonal of a rectangle is 10

cm. What is the number of square centimeters in

the maximum possible area of this rectangle?

Answer 50 (square centimeters)

70. If x y 5 , x z 8 and y z 1 ,

what is the value of x y z ?

Answer 7

71. Simplify, assuming that and

Answer

72.What is the arithmetic mean of the first ten

numbers in the arithmetic sequence 5, 11,

17, 23, ... ?

Answer 32

73. Billy and Bobbi each selected a positive

integer less than 200. Billys number is a

multiple of 18, and Bobbis number is a multiple

of 24. What is the probability that they

selected the same number? Express your answer as

a common fraction.

Answer

74. If pens can be purchased for 35 each or

three for 1.00, how many dollars are saved if 63

pens are purchased in sets of three instead of

individually? Express your answer to the nearest

hundredth.

Answer 1.05 (dollars)

75. What is the units digit in the product of all

natural numbers from 1 to 99, inclusive?

Answer 0

76. How many feet longer is 1 of a mile than 60

of a yard?

Answer 51 (feet)

77. If of 230 is 4x , then what is the

value of x ?

Answer 14

78. Ervina made 62.5 of the shots she took. If

she made 25 shots, then how many shots did

she take?

Answer 40 (shots)

79. The lengths, in centimeters, of the sides of

a right triangle are integers, and the sum of the

shortest and longest sides is 18 cm. How

many centimeters are in the perimeter of the

triangle?

Answer 30 (centimeters)

80. When n is divided by 3, the remainder is 2.

What is the remainder when 5n is divided by 3?

Answer 1