MATHCOUNTS

1
MATHCOUNTS
1999 State Competition Countdown Round
2
How many of the first 100 positive integers are
neither perfect squares nor perfect cubes?
3
Answer 88 (integers)
4
A palindrome is a number that reads the same
forwards and backwards. How many palindromes are
between 1000 and 10,000?
5
Answer 90 (palindromes)
6
The surface of a 4 x 5 x 6 block is painted
yellow. If the block is separated into one-inch
cubes, what is the number of cubes with exactly
one yellow face?
7
Answer 52 (cubes)
8
The lengths of two sides of a right triangle are
meters and meters. The number
of meters in the length of the third side is a
whole number. How many meters are in the third
side of the triangle?
9
Answer 8 (meters)
10
A bag contains yellow and blue marbles. The
probability of selecting a yellow marble from the
bag is . If there are 27 blue marbles in the
bag, what is the number of yellow marbles?
11
Answer 24 (marbles)
12
A uniform piece of lumber is 12 ft long and
weighs 30 lb. If a 5 ft length is removed, what
is the number of pounds in the remaining piece?
Express your answer as a mixed number.
13
Answer 17 (pounds)
14
If q, r and s each represent a different integer
0-9, what is the value of s?
15
Answer 8
16
Compute .
17
Answer 10,000
18
What number is 11 of 13 239?
19
Answer 1471
20
For what whole number n is
?
21
Answer 5
22
What is the ratio of 0.16 to 0.83? Express your
answer as a common fraction.
23
Answer
24
The measure of the angles of a pentagon are in
the ratio of 33345. What is the number of
degrees in the measure of the largest angle?
25
Answer 150 (degrees)
26
A digital clock shows hours and minutes. How
many times during a 12-hour period will the sum
of the digits on this clock be 3?
27
Answer 12 (times)
28
It takes 6 days for a team of 6 workers to
carpet a house. At the same rate, what is the
minimum number of additional workers needed to
carpet the same house in just 5 days?
29
Answer 2 (workers)
30
If and , what is
the value of ?
31
Answer 114
32
John has some loaves of bread in a bag. If he
had 6 more loaves, he would have three times as
many. How many loaves does John have in the bag?
33
Answer 3 (loaves)
34
If a two-digit prime number is selected at
random, what is the probability that the sum of
its digits is 9?
35
Answer 0
36
In right triangle ABC, CAB is a right angle.
Point M is the midpoint of BC. What is the
number of centimeters in the length of median AM?
Express your answer as a decimal to the nearest
tenth.
37
Answer 2.5 (centimeters)
38
What is the number of degrees in the exterior
angle of the largest angle of a triangle if the
interior angles are in the ratio of 235?
39
Answer 90 (degrees)
40
A 12 feet by 8 feet rectangular Persian rug is
centered within a rectangular room so that a
region of constant width surrounds the carpet.
Given that 96 square feet of flooring is not
covered by the rug, what is the number of feet in
the measure of the longer dimension of the room?
41
Answer 16 (feet)
42
What is the least positive integer that is not a
factor of 7! ?
43
Answer 11
44
A right circular cone is inscribed in a right
circular cylinder. The volume of the cylinder is
72p cubic centimeters. What is the number of
cubic centimeters in the space inside the
cylinder but outside the cone? Express your
answer in terms of p.
45
Answer 48p (cubic centimeters)
46
Compute 12 of 240.
47
Answer 30
48
In an isosceles trapezoid, the lengths of the
bases are 8 cm and 16 cm. The length of a
diagonal is 13 cm. What is the number of square
centimeters in the area of the trapezoid?
49
Answer 60 (square centimeters)
50
What is the greatest two-digit whole number that
can be expressed as the product of two
consecutive whole numbers?
51
Answer 90
52
What is the greatest three-digit number that is
divisible by 6?
53
Answer 996
54
If the measure of each side of a square is
decreased by 10, by what percent is the area
decreased?
55
Answer 19 (percent)
56
What is the units digit of ?
57
Answer 0
58
All squares have the same value, and all circles
have the same value. What is the value of the sum
of three circles?
21
19
?
59
Answer 9
60
Six less than a number is multiplied by 6. Eight
less than the same number is multiplied by 8.
The two results are equal. What is the number?
61
Answer 14
62
When Sarah turns 36 years old, her father will be
62. Currently, Sarah is of her fathers age.
How many years old is Sarah now?
63
Answer 13 (years)
64
The sum of seven consecutive even numbers is
13,902. What is the smallest of the seven
numbers?
65
Answer 1980
66
Eleven coins total 1.19, but no combination of
the coins will give a value of exactly 1. How
many of the coins are quarters?
67
Answer 3 (quarters)
68
Jimi is taking a bicycle trip and needs to arrive
by 6 oclock. If he travels at an average rate
of 15 mph, hell arrive at his destination one
hour early, but if he travels at an average rate
of 10 mph, hell arrive one hour late. How many
miles away is his destination?
69
Answer 60 (miles)
70
Mark baked cookies for the neighborhood children.
He gave each child 6 cookies, and he had 7
cookies remaining. He was going to give another
cookie to each child, but he was 1 cookie short.
How many cookies had Mark baked?
71
Answer 55 (cookies)
72
An eight-digit number contains two 5s, two 6s,
two 7s and two 8s. The 5s are separated by
one digit, the 6s are separated by two digits,
the 7s are separated by three digits, and the
8s are separated by four digits. What is the
greatest possible value for the number?
73
Answer 85,756,876
74
Calculate .
75
Answer 27
76
Calculate and express your answer as a common
fraction
77
Answer
78
What is the least positive integer n such that
?
79
Answer 5
80
How many different three-member committees can
be chosen from a group of ten people?
81
Answer 120 (committees)
82
A rectangle with length 8 units and width 5 units
is centered at the origin of a coordinate plane,
and its longer side is parallel to the x-axis.
What percent of the rectangular region is to the
right of the line with equation ?
83
Answer 75 (percent)
84
What is the smallest natural number that has
exactly five factors?
85
Answer 16
86
For what value of n does ?
87
Answer 2
88
What is the sum of the real values of x for which
the following expression is undefined? Express
your answer as a common fraction.
89
Answer
90
A 3 x 5 photograph can be enlarged to a
panoramic view that is 5 x 10. What is the
ratio of the area of the original photograph to
the area of the enlargement? Express your answer
as a common fraction.
91
Answer
92
The sum of two numbers is 9, and their difference
is 7. What is the positive difference of the
squares of these two numbers?
93
Answer 63
94
One diagonal of a rhombus measures 48 inches.
Given that one side of the rhombus has length 26
inches, what is the number of square inches in
the area of the rhombus?
95
Answer 480 (square inches)
96
A centipede climbs a 40-foot tree. Each day he
climbs 5 feet, and each night he slides down 2
feet. In how many days will the centipede reach
the top of the tree?
97
Answer 13 (days)
98
For how many integers n is n - 5 lt 3 ?
99
Answer 5 (integers)
100
Bill, Frank and Grete are playing a game. Bill
is three times as likely to win the game as
Frank, and Frank is two times as likely to win
the game as Grete. Assuming no ties, what is the
probability that Grete will win the game?
Express your answer as a common fraction.
101
Answer
102
What is the greatest rational number n such that
?
103
Answer 8
104
What is the sum of the reciprocals of 1, 2, 3, 4
and 5? Express your answer as a common fraction.
105
Answer
106
A jar contains 60 marbles, each colored either
green or yellow, all the same size. There are
twice as many green marbles as yellow in the jar.
How many yellow marbles must be added to the jar
so the probability of choosing a green marble is
?
107
Answer 20 (marbles)
108
Each of the 30 students in a physical education
class plays the same number of minutes of
full-court basketball in a class period. Ten
students play at a time, and the class period
lasts 51 minutes. What is the number of minutes
each student plays?
109
Answer 17 (minutes)
110
A space diagonal of a given cube is
inches long. What is the number of square inches
in the surface area of the cube?
111
Answer 600 (square inches)
112
A monument is constructed so that the floor is a
regular hexagon with sides measuring 8 feet.
What is the number of square feet in the area of
the floor? Express your answer in simplest
radical form.
113
Answer (square feet)
114
In China, small spheres are used to exercise a
persons hands. If the diameter of one of these
spheres measures 2 inches, how many square inches
are in its surface area? Express your answer in
terms of p.
115
Answer 4p (square inches)
116
For what value of n does
?
117
Answer 10
118
Sam and Janet each have a whole number of
dollars, and of Sams money equals of
Janets money. The sum of their money is more
than 10. What is the least whole number of
dollars they could have combined?
119
Answer 15 (dollars)
120
What is the greatest digit that can be placed in
the box so that the six-digit number shown is
divisible by 4?
352,9 2
121
Answer 9
122
When you roll three number cubes, each numbered 1
through 6, what is the probability that the tops
of the three cubes will show either all even
numbers or all odd numbers? Express your answer
as a common fraction.
123
Answer
124
For every 3 herks, there are 4 merks. Together,
there are 84 herks and merks. How many more
merks than herks are there?
125
Answer 12 (merks)
126
Given that , evaluate
. Express your answer as a decimal to the
nearest hundredth.
127
Answer 0.25
128
Mark earns 8.00 per hour for the first 8 hours
he works each day. If he works more than 8
hours, he gets paid 1.5 times his hourly rate for
the extra hours. On Monday, Mark worked from
730 AM to 630 PM and took an hour for lunch.
If his lunch hour is unpaid, how many dollars did
Mark earn on Monday?
129
Answer 88 (dollars)
130
If three values from the list below are selected
at random without replacement, what is the
probability that they are equal? Express your
answer as a common fraction.
40 of 1 0.4
131
Answer
132
Points A, B, C, D and E are five points on a line
segment with endpoints A and E. The points are
in the order listed above from left to right. CD
AB, BC CD, AB AE, and AE 12
cm. What is the number of centimeters in the
length of AD? Express your answer as a decimal
to the nearest tenth.
133
Answer 10.5 (centimeters)
134
What is the maximum possible value of x in the
inequality ?
135
Answer 8
136
Three straight lines intersect at O, and OD
bisects COE. The ratio of COB to BOF
is 72. What is the number of degrees in m
COD?
137
Answer 20 (degrees)
138
Jena starts on a trip at 800 AM and travels at a
rate of 50 mph. Harvey starts out at 1030 AM
the same day and travels the same route at a rate
of 75 mph. What is the sum of the total miles
traveled by Harvey and Jena when Harvey catches
up with Jena?
139
Answer 750 (miles)
140
A right circular cylinder with radius 2 inches
and height 1 inch has the same volume as a second
right circular cylinder. The radius of the
second cylinder is 1 inch. What is the number of
inches in the second cylinders height?
141
Answer 4 (inches)
142
For how many values of x does
?
143
Answer 2 (values)
144
Using only quarters, dimes, nickels, and pennies,
what is the minimum number of coins needed to get
91 cents?
145
Answer 6 (coins)
146
The measure of an exterior angle of a triangle is
105o. If one of the non-adjacent interior angles
measures 38o, what is the number of degrees in
the other non-adjacent interior angle?
147
Answer 67 (degrees)
148
Given that 36 feet of fencing are needed to fence
the sides of a triangular region with sides in
the ratio 345, how many square feet are in the
area of the triangular plot?
149
Answer 54 (square feet)
150
Larry traveled from his house to the mall at 40
mph and returned along the same route at 60 mph.
What was his average speed in miles per hour for
the entire trip?
151
Answer 48 (miles per hour)
152
How many seconds does it take the second hand of
a clock to rotate through an angle of 36o?
153
Answer 6 (seconds)