1' What is the sum of the number of faces, vertices and edges in a triangular prism

1
MATHCOUNTS?
2003 State Competition Countdown Round
2
1. Compute the product
3
Answer 16
4
2. What is the least integer greater than
?
5
Answer 18
6
3. A number is randomly selected from the first
20 positive integers. What is the probability
that it is divisible by 2, 3 or 4? Express your
answer as a common fraction.
7
Answer
8
4. A tree is now 12 feet tall and grows 18 inches
a year. In how many years will the tree be 36
feet tall?
9
Answer 16 (years)
10
5. Stacey is standing in a field. She walks 11
meters west, 30 meters north, 4 meters
west and finally 22 meters south. How many
meters is she from her starting point?
11
Answer 17 (meters)
12
6. What is the remainder when 2003 is divided by
11?
13
Answer 1
14
7. Use each of the digits 3, 4, 6, 8 and 9
exactly once to create the greatest possible
five-digit multiple of 6. What is that multiple
of 6?
15
Answer 98,634
16
8. The sum of the first and third of three
consecutive integers is 118. What is the value
of the second integer?
17
Answer 59
18
9. A palindrome is a number that reads the same
forward and backward, such as 1331. What is the
palindrome between 500 and 750 that is divisible
by 3 and whose digits have a product of 200?
19
Answer 585
20
10. If 2ab 12, evaluate 8a2b2.
21
Answer 288
22
11. A candy bar weighs 4.48 ounces. A person
eats half of the bar with the first bite. If a
person eats half of the amount remaining with
each subsequent bite, how many bites have been
taken when exactly 0.07 ounces remain?
23
Answer 6 (bites)
24
12. Each corner of a cube is cut off to leave an
equilateral triangle as shown. How many faces
does the new polyhedron have?
25
Answer 14 (faces)
26
13. Everyone in a class of 30 students
takes math and history. Seven students received
an A in history, 13 received an A in math and
four received an A in both courses. How many
students did not receive an A in either course?
27
Answer 14 (students)
28
14. What is the eighth term in thearithmetic
sequence Express your answer is simplestform.
29
Answer 3
30
15. The diagonals of a regular hexagon have two
possible lengths. What is the ratio of the
shorter length to the longer length? Express
your answer as a common fraction in simplest
radical form.
31
Answer
32
16. What is the greatest integer value of n such
that 3n lt 10,000?
33
Answer 8
34
17. What is the units digit of the product of the
first 100 prime numbers?
35
Answer 0
36
18. Solve for m
37
Answer 6
38
19. A regular pentagon is rotated
counterclockwise about its center. What is the
minimum number of degrees it must be rotated
until it coincides with its original position?
39
Answer 72 (degrees)
40
20. Which integer is closest to the cube root of
100?
41
Answer 5
42
21. Dora has three candles, which will burn for
3, 4 and 5 hours, respectively. If she
wants to keep two candles burning at all times,
what is the greatest number of hours she can burn
the candles?
43
Answer 6 (hours)
44
22. What is the expression for the area of the
circle which has a circumference of C? Express
your answer as a rational expression in terms of
C and .
45
Answer (or equivalent)
46
23. Mary completes a 15-mile race in 2.5 hours.
What is her average speed in miles per hour?
47
Answer 6 (miles per hour)
48
24. If 15 of N is 45 of 2003, what is the value
of N?
49
Answer 6009
50
25. How many digits are in the whole-number
representation of 10100 9100?
51
Answer 100 (digits)
52
26. One night the temperature fell 3 degrees
every hour from 6 p.m. through midnight. The
temperature was 17 degrees at 6 p.m. What was
the temperature at midnight that night?
53
Answer 1 (degrees)
54
27. How many square units are in the area of the
largest square that can be inscribed in a circle
with radius 1 unit?
55
Answer 2 (square units)
56
28. Express the sum as a common fraction .1
.02 .003 .0004 .00005.
57
Answer
58
29. Compute . Expressyour answer in
simplest form.
59
Answer 12
60
30. What is the 2003rd term of the sequence of
odd numbers 1, 3, 5, 7, ?
61
Answer 4005
62
31. For how many different digits n is the
two-digit number 6n divisible by n?
63
Answer 6 (digits)
64
32. If 1 skip equals 4 hops, and 1 jump equals 3
skips, what is the total number of hops in a hop,
a skip and a jump?
65
Answer 17 (hops)
66
33. Calvin is 150 cm tall, which is 75 of
Darryls height. How many centimeters tall is
Darryl?
67
Answer 200 (centimeters)
68
34. An edge of a square can be expressed as 4x
15 meters or as 20 3x meters. What is its area
in square meters?
69
Answer 25 (square meters)
70
35. For what integer x is ?
71
Answer 5
72
36. One caterer charges a basic fee of 100 plus
15 per person. A second caterer charges a basic
fee of 200 plus 12 per person. What is the
least number of people for which the second
caterer is cheaper?
73
Answer 34 (people)
74
37. Compute
75
Answer 120
76
38. What percent of 16 is 50 of 50 of 48?
77
Answer 75 (percent)
78
39. Solve for n
79
Answer 72
80
40. Let GCF (a, b) denote the greatest common
factor of a and b and let LCM (a, b) denote the
least common multiple of a and b. What is LCM
(GCF (24, 32), GCF (12, 18))?
81
Answer 24
82
41. The areas of squares A1 and A2 are 25 square
centimeters and 49 square centimeters
respectively. What is the number of square
centimeters in the area of rectangle A3?
A2
A3
A1
83
Answer 35 (square centimeters)
84
42. The square with vertices (1, 1), (1,
1), (1, 1) and (1, 1) is cut by the line
into a triangle and a pentagon.
What is the number of square units in the area of
the pentagon? Express your answer as a decimal
to the nearest hundredth.
85
Answer 3.75 (square units)
86
43. A horse 24 feet from the center of a
merry-go-round makes 32 revolutions. In order to
travel the same distance, how many revolutions
would a horse 8 feet from the center have to
make?
87
Answer 96 (revolutions)
88
44. A digital clock reads 830. What will it
read in 100 minutes?
89
Answer 1010
90
45. How many integers n satisfy (n 2)(n 4) lt
0?
91
Answer 5 (integers)
92
46. A toy store manager received a large order of
Mr. Slinkums just in time for the holidays. The
manager places 20 of them on the shelves,
leaving the other 120 Mr. Slinkums in storage.
How many Mr. Slinkums were in this order?
93
Answer 150 (Mr. Slinkums)
94
47. How many three-digit positive integers exist,
all of whose digits are 2s and/or 5s?
95
Answer 8 (integers)
96
48. Define N by the formula N .5(N) 1.
Calculate ((50)).
97
Answer 8
98
49. What is the 100th term of the arithmetic
sequence 6, 10, 14, 18, ?
99
Answer 402
100
50. On the given cube, M is the midpoint of
segment and N is the midpoint of segment
. The plane MNEF slices the solid cube
into two polyhedra. The volume of the smaller
polyhedron is what common fraction of the
volume of the whole cube?
E
F
B
A
M
N
D
C
101
Answer
102
51. A baby blue whale weighed 3 tons at birth.
When it was 10 days old, it weighed 4 tons.
If its rate of growth was constant for the first
60 days of its life, how many tons did this whale
weigh when it was 45 days old? Express your
answer as a decimal to the nearest tenth.
103
Answer 7.5 (tons)
104
52. What is the median of the first ten positive
integers? Express your answer as a decimal to
the nearest tenth.
105
Answer 5.5
106
53. The length of each edge of a cube is
decreased by 40. By what percent does the
volume of the cube decrease? Express your answer
to the nearest tenth.
107
Answer 78.4 (percent)
108
54. Assuming that the birth of a boy or a girl is
equally likely, what is the probability that the
three children in a family include at least one
boy and one girl? Express your answer as a
common fraction.
109
Answer
110
55. Two different numbers are randomly selected
from the set 1, 2, 3, 4 and they are
multiplied. What is the probability that the
product is even? Express your answer as a common
fraction.
111
Answer
112
56. The cost of a meal at a restaurant was 15
before tax and tip. If the 7 tax and an 18 tip
are each based solely upon the cost of the meal,
what is the total cost in dollars of the meal,
tax and tip? Express your answer as a decimal to
the nearest hundredth.
113
Answer 18.75 (dollars)
114
57. A diagonal of a polygon is a segment joining
two non-consecutive vertices of the
polygon. How many diagonals does a regular
octagon have?
115
Answer 20 (diagonals)
116
58. Let S be the set of all three-digit
numbers formed by three consecutive digits in
increasing order. What is the greatest common
factor of all the three-digit numbers in S?
117
Answer 3
118
59. To take quizzes, each of 30 students in
a class is paired with another student. If the
pairing is done randomly, what is the probability
that Margo is paired with her best friend, Irma?
Express your answer as a common fraction.
119
Answer
120
60. What is the value of the number which lies
of the distance from 98.36 to 100.25 on a number
line? Express your answer as a decimal to the
nearest hundredth.
121
Answer 99.62
122
61. What is the units digit of 2323?
123
Answer 7
124
62. In a recent year, the number of residents of
Palm Springs was 45,000 and there were 3 million
visitors to Palm Springs. What was the average
number of visitors per resident of Palm Springs?
Express your answer to the nearest whole number.
125
Answer 67 (visitors)
126
63. A couple has the opportunity to choose three
of five concerts for the coming season. In how
many ways can they choose the three concerts?
127
Answer 10 (ways)
128
64. If 4 horses eat 4 bales of hay in 4 days, how
many days will it take 20 horses to eat 30 bales
of hay?
129
Answer 6 (days)
130
65. What is the sum of all positive two-digit odd
integers?
131
Answer 2475
132
66. Two numbers are in the ratio of 56. Their
sum is 77. What is the positive difference
between the two numbers?
133
Answer 7
134
67. Each successive term in the sequence 2048,
512, 128, x, y, 2, is obtained by multiplying
the previous term by a constant. What is the
value of x y?
135
Answer 40
136
68. Karen needs pieces of lace each yards
long. How many whole pieces can she cut from a
9-yard piece of lace?
137
Answer 13 (pieces)
138
69. The side lengths of a triangle are 14 cm, 48
cm and 50 cm. How many square centimeters are in
the area of the triangle?
139
Answer 336 (square centimeters)
140
70. Which of the following is the highest annual
wage 600 per week, 2500 per month or 31,000
per year?
141
Answer 600 (dollars per week)
142
71. What is the remainder when the sum 12 22
32 102 is divided by 11?
143
Answer 0
144
72. What is the greatest integer value of n such
that 25! is divisible by 10n?
145
Answer 6
146
73. The capacity of a school is 1100 students and
the current enrollment is 980 students. If the
student population increases at a rate of 5
percent per year, in how many years will the
enrollment exceed the capacity?
147
Answer 3 (years)
148
74. A rectangle has a perimeter of 30 units and
its dimensions are whole numbers. What is the
maximum possible area of the rectangle in square
units?
149
Answer 56 (square units)
150
75. A yogurt shop sells four flavors of yogurt
and has six different toppings. How many
combinations of one flavor and two different
toppings are available?
151
Answer 60 (combinations)
152
76. What is the units digit of the sum of the
nine terms of the sequence 1! 1, 2! 2,
3! 3, , 8! 8, 9! 9 ?
153
Answer 8
154
77. Ashley and Chris each prepare a guest list
for a party. No name appears on both lists. Half
of Chris guests attend and a third of Ashleys
guests attend for a total of 100 guests. Ashleys
list contained 180 names. How many names were on
Chris list?
155
Answer 80 (names)
156
78. If f (x) 3x2 5, what is the value of f (
f (1)) ?
157
Answer 7
158
79. When rolling two fair, standard six-faced
dice, what is the probability of rolling the same
number on both dice? Express your answer as a
common fraction.
159
Answer
160
80. Simplify
161
Answer 33