MATHCOUNTS

1
MATHCOUNTS

• 1990-1991
• State Countdown Round

2
How many of the even counting numbers less than
100 are prime?
3
1
4
The probability of rain on any given day in
Atlanta is 20. After how many days would you
expect it to have rained on 30 days?
5
150 (days)
6
Two fair tetrahedral (four-sided) dice are
thrown. If the faces of each are numbered 3, 5,
7, and 9, what is the probability that the sum of
the numbers on the bottom faces is a prime?
7
0
8
What is the shortest distance that can be
traveled if you start at any point, A, B, C, or D
and visit the other three points once?
A 6 5 D 3 B 6 4
5 C
9
13
10
How many positive integer pairs (x, y) satisfy A
x y, 3when A 2?
11
5 (pairs)
12
If the digit 7 is placed to the right of a
three-digit number, forming a four-digit number,
the value of the original number increases by
7,000. What was the original number?
13
777
14
Fifty cards numbered 1 to 50 are placed in a box.
If a card is selected at random, what is the
probability that the card is a prime number and a
multiple of seven? Express your answer as a
common fraction.
15
1 50
16
What is the last digit in the product of all
natural numbers between 1 and 100?
17
0
18
If 5! is expressed as the product of its prime
factors in exponential form, what is the sum of
its exponents?
19
5
20
This table shows the distribution of scores of
Mr. Sampsons 10-point quiz. What is the median
score?
Test Scores Frequency
10 I
9 II
8 III
7 IIII
6 IIII
5
4 I
3
2 II
21
7
22
A certain type of 8 ½ by 11 paper has a
thickness of 0.02 cm. If 15,000 sheets of paper
are stacked on top of each other, what is the
height, in meters, of the stack?
23
3 (meters)
24
Points A, B, C, and D are located on AB such that
AB 3AD 6BC. If a point is selected at
random, what is the probability that it is
between C and D? Express your answer as a common
fraction.A D C B
25
1 2
26
Three identical squares are placed side by side
to form a rectangle with a perimeter of 104
inches. What is the area, in square inches, of
each square?
27
169 (sq. in.)
28
Brad is less than 30 years old. His age is a
multiple of 5. Next year his age will be a
multiple of 7. How old is Brad now?
29
20 (years old)
30
How many different triangles are in the figure?
31
13 (triangles)
32
How many even numbers are between 202 and 405?
33
101
34
What is the positive square root of 16x16?
35
4x8
36
When n is divided by 5, the remainder is 1. What
is the remainder when 3n is divided by 5?
37
3
38
Express 10 of 30 of 50 as a decimal.
39
1.5
40
A 1, 2, 3, 4, 5, 6, 7, 8, 9, . What
fractional part of the numbers in set A are not
divisible by 3?
41
2 3
42
Six softball teams are to play each other once.
How many games are needed?
43
15 (games)
44
Two ropes, 18 meters in length and 24 meters in
length, need to be cut into pieces which are all
the same length. What is the greatest possible
length of each piece?
45
6 (meters)
46
What value of n makes this sentence true? n
n 1 6 9
47
2
48
If f(x) 3x 2 and g(x) (x 1)2, what is
f(g(-2))?
49
29
50
If March 1 is a Monday, what day of the week will
it be 270 days later?
51
Friday
52
Given x 3 and y 2, simplify 2x3 3y2 6
53
7
54
Simplify 7! 4!3!
55
35
56
In how many different orders can a group of six
people be seated around a round table?
57
120 (ways)
58
How many different arrangements are there using
the letters in the word PARALLEL?
59
3,360 (arrangements)
60
What is v200 to the nearest tenth?
61
14.1
62
What is the positive geometric mean of 8 and 18?
63
12
64
Find the midpoint of the line segment with
endpoints (5, -5) and ( -5, 5).
65
(0, 0)
66
Give the decimal expression for q of q.
67
0.01q2
68
Solve for x 2x 2 128
69
5
70
The area of a triangle is 600 square feet. Find
the altitude, in feet, of the triangle if the
length of the base is 30 feet.
71
40 (meters)
72
The longest side of a right triangle is 5 feet
and the shortest side is 3 feet. What is the
area of the triangle in square meters?
73
6 (sq. meters)
74
If each side of a square is increased by 50,
what is the percent of increase in the area of
the square?
75
125
76
Four lines are drawn in a plane. What is the
maximum number of distinct regions in which the
lines could divide the plane?
77
11
78
Simplify .6 .3 .9(all numbers are
repeating decimals)
79
1
80
What percent of 80 of a number will lave the
number unchanged?
81
125
82
Find the number of square units in the area of
the shaded region (0, 30) (20, 30)
(30, 30) (30, 20) (0,
10) (0, 0) (10, 0) (30, 0)
83
500 (sq. units)
84
If y -7 v(5 x), find x when y 3
85
- 95
86
In quadrilateral ABCD, angle BAD and angle CDA
are trisected as shown. What is the degree
measure of angle AFD? B C
F x E y A x X y
y D
87
80 (degrees)
88
Find the number of square units in the area of
the triangle. (-2, 6) (-6,
2) x (0,
0) y
89
32 (sq. units)
90
What is the smallest integer that satisfies 2x
3lt 8?
91
- 2
92
How many two-digit whole numbers are divisible by
14?
93
7
94
What percent of the whole numbers less than 100
have no remainders when divided by 5?
95
20 ()
96
What is the 100th odd whole number?
97
199
98
ABCD is a square 4 inches on a side, and each of
the inside squares is formed by joining the
midpoints of the outer squares sides. What is
the area of the shaded region?
99
4 (sq. inches)
100
If all multiples of 3 and all multiples of 4 are
removed from the list of numbers 1 through 100,
how many whole numbers are left?
101
50
102
Divide 4abc by 2a2b 3d2
103
6cd2a
104
What is the simplest radical form of 3v4 ? 6v2
105
v2
106
Two fair cubical dice are tossed. What is the
probability, expressed as a common fraction, that
the sum of the numbers showing on the dice will
be four?
107
1 12
108
If ab is defined as a b, 2what is
the value of 6(35)?
109
5
110
If x and y are positive integers such that xy
108, what is the least possible value of x y?
111
21
112
Find vv256,000
113
40
114
Express the value of 1 of 10 of 0.001 in
scientific notation. 100
115
1 or(1.0) X 10-6
116
Find the sum of all solutions to this
equationx2 62 102
117
0
118
What is the sum of the lengths, in centimeters,
of the two legs of a 30-60-90 right triangle, if
the length of the hypotenuse is 2v6 cm.?
119
v6 3 v2
120
How many diagonals can be drawn for a hexagon?
121
9 (diagonals)
122
FIN!