1. What is the slope of the line determined by any two solutions to the equation ? Express your answer as a common fraction.

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MATHCOUNTS?
2002 State Competition Countdown Round
2
1. What is the slope of the line determined by
any two solutions to the equation
? Express your answer as a common fraction.
3
Answer
4
2. How many different prime factors does 20!
have?
5
Answer 8 (factors)
6
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?
7
Answer 40 (cents)
8
4. Compute
9
Answer
10
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?
11
Answer 202 (units)
12
6. What is the sum of the roots of 2x2 1 3x
? Express your answer as a common fraction.
13
Answer
14
7. The circumference of a circle increases from
20 cm to 30 cm. By how many centimeters will
the radius increase?
15
Answer 5 (centimeters)
16
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
17
Answer 4.8 (centimeters)
18
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?
19
Answer 150 (limes)
20
10. Solve for x
21
Answer
22
11. What is the length of the diagonal of a
square with side length cm? Express
your answer in simplest form.
23
Answer 100 (centimeters)
24
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?
25
Answer 24 (inches)
26
13. Compute 25 (1999 2000 2001 2002).
27
Answer 200,050
28
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
29
Answer 16 (squares)
30
15. For what integer n is
31
Answer 6
32
16. If 3 of 400 is equal to 5n 2, then
what is the value of n ?
33
Answer 2
34
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).
35
Answer 108 (square units)
36
18. Determine the sum of all single-digit
replacements for n such that the number
42,789,n37 is divisible by 3.
37
Answer 15
38
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?
39
Answer 125 (unit cubes)
40
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?
41
Answer 343 (pieces)
42
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?
43
Answer 11 (integers)
44
22. For what value of b is the set
equal to the set 4, 5, 6?
45
Answer 4
46
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.
47
Answer
48
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.
49
Answer
50
25. How many miles will a race car travel in 90
seconds at a rate of 80 miles per hour?
51
Answer 2 (miles)
52
26. Compute
53
Answer 60
54
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.
55
Answer
56
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?
57
Answer 72 (miles)
58
29. What is the largest prime factor of 2323?
59
Answer 101
60
30. What is 10 of 20 of 30 of 10,000 ?
61
Answer 60
62
31. How many factors of are perfect
squares?
63
Answer 12 (factors)
64
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.
65
Answer 90 (square units)
66
33. What is the least four-digit whole number
that is both a perfect square and a perfect cube?
67
Answer 4096
68
34. How many non-empty subsets containing only
prime numbers does the set 1, 2, 3, 4, 5, 6, 7
have?
69
Answer 15 (subsets)
70
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
1999Aluminum 37 tons 88 tons
Tin 117 tons 184 tons Glass
801 tons 353 tons Newspaper
1664 tons 3726 tons Plastic 93
tons 266 tons
71
Answer 3 (products)
72
36. If what is the value
of x ?
73
Answer 6
74
37. When is expressed as arepeating
decimal of the form 0.ababab..., what is the
product of a and b ?
75
Answer 5
76
38. What decimal number isexactly of the
way from .826to .827 ?
77
Answer .82675
78
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.
79
Answer 35.65 (dollars)
80
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?
81
Answer 6 (square inches)
82
41. Lon doubled a four-digit number and then
subtracted 1275. The result was 3573. What
four-digit number did Lon double?
83
Answer 2424
84
42. How many minutes are in 10 of a 24-hour day?
85
Answer 144 (minutes)
86
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?
87
Answer 4 (rows)
88
44. If a triangle has two sides of lengths 5 and
7 units, then how many different integer lengths
can the third side be?
89
Answer 9 (lengths)
90
45. What is the least perfect square with 3
different prime factors?
91
Answer 900
92
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.
93
Answer
94
47. How many of the following numbers are greater
than 3 and less than 4?
95
Answer 1 (number)
96
48. The sum of ten consecutive odd numbers is
640. What is the greatest of these ten numbers?
97
Answer 73
98
49. Two integers have a sum of and a
product of 54. What is the smaller of these two
integers?
99
Answer
100
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 ?
101
Answer 97,610
102
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.
103
Answer
104
52. Each of six softball teams is to play each
other team once. How many games are needed?
105
Answer 15 (games)
106
53. The endpoints of a line segment are (2, 3)
and (8, 15). What is the sum of the coordinates
of the midpoint?
107
Answer 14
108
54. What is the probability that a
randomly-selected two-digit number is a multiple
of 14? Express your answer as a common fraction.
109
Answer
110
55. Nine different two-digit numbers can be
formed with the digits 1, 3 and 7. How many of
these numbers are prime?
111
Answer 7 (numbers)
112
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?
113
Answer 7 (marbles)
114
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?
115
Answer 169 (square inches)
116
58. How many of the first 500 positive
integers are divisible by 3, 4 and 5?
117
Answer 8 (integers)
118
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?
119
Answer 4 (square units)
120
60. Hillary has eleven coins, all dimes and
nickels. In total, the coins are worth 75 cents.
How many nickels does she have?
121
Answer 7 (nickels)
122
61. What is the reciprocal of ?
Express your answer in simplest form.
123
Answer 45
124
62. How many even numbers are greater than 202
and less than 405?
125
Answer 101 (numbers)
126
63. Compute
127
Answer 210
128
64. What percent of the integers from 1 to 100,
inclusive, leave remainder 1 when
divided by 5?
129
Answer 20 (percent)
130
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?
131
Answer 144 (area codes)
132
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.
133
Answer
134
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?
135
Answer 22
136
68. What is the sum of all positive integer
divisors of 77?
137
Answer 96
138
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?
139
Answer 50 (square centimeters)
140
70. If x y 5 , x z 8 and y z 1 ,
what is the value of x y z ?
141
Answer 7
142
71. Simplify, assuming that and

143
Answer
144
72.What is the arithmetic mean of the first ten
numbers in the arithmetic sequence 5, 11,
17, 23, ... ?
145
Answer 32
146
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.
147
Answer
148
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.
149
Answer 1.05 (dollars)
150
75. What is the units digit in the product of all
natural numbers from 1 to 99, inclusive?
151
Answer 0
152
76. How many feet longer is 1 of a mile than 60
of a yard?
153
Answer 51 (feet)
154
77. If of 230 is 4x , then what is the
value of x ?
155
Answer 14
156
78. Ervina made 62.5 of the shots she took. If
she made 25 shots, then how many shots did
she take?
157
Answer 40 (shots)
158
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?
159
Answer 30 (centimeters)
160
80. When n is divided by 3, the remainder is 2.
What is the remainder when 5n is divided by 3?
161
Answer 1