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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 2  3  0 ?
x y
Express your answer as a
common fraction.
3
Answer: 
2
2. How many different prime
factors does 20! have?
Answer: 8 (factors)
3. Murphy’s sells notebooks at a
cost of five for $2, Right-Ade
sells six for $3 and Callahan’s
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:
1  2  3  4  5  ...  99  100 .
Answer: 50
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.
3
Answer:
2
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. ABC is similar to DEF .
What is the number of
centimeters in the length of
EF ? Express your answer as a
decimal to the nearest tenth.
B
8cm
C
5cm
E
A
F
3cm
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: 2
x 3
4
x1
.
Answer:  5
11. What is the length of the
diagonal of a square with side
length 50 2 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
(10  1) (10  1)  999,999,999,999 ?
n
n
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 cube’s 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 Lola’s 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
{2b 3, b, b  2} equal to the set
{4, 5, 6}?
Answer: 4
23. Kristin has 5 different pairs
of earrings. What is the
probability that two randomlyselected earrings will be a
matching pair? Express your
answer as a common fraction.
1
Answer:
9
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:
3
4
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:
2 2 2 2
.
4
4
5
6
7
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:
2
5
28. A snow plow clears 16 miles
of roads per hour. If the plow’s
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 2  3
are perfect squares?
5
6
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 regarding residential recycling in Tallahassee, 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)
3
36. If
x

5
x
1
 1 3 , what
is the value of x ?
Answer: 6
5
37. When
is expressed as a
33
repeating decimal of the form
0.ababab..., what is the product
of a and b ?
Answer: 5
38. What decimal number is
3
exactly 4 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.
3
Answer:
10
47. How many of the following
numbers are greater than 3 and
less than 4?
3
3
7
8
3
25
7
20
3
2
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
21 and a product of 54. What is
the smaller of these two integers?
Answer: 18
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.
8
Answer:
3
25
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:
7
90
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
0.02 ? Express your answer in
simplest form.
Answer: 45
62. How many even numbers are
greater than 202 and less than
405?
Answer: 101 (numbers)
7!
63. Compute:
.
3! 2 ! 2 !
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:
1
2
67. Mr. Sanchez’s 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 ab  0
and a  b :
11
a b
.
11
b a
Answer: 1
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.
Billy’s number is a multiple of
18, and Bobbi’s number is a
multiple of 24. What is the
probability that they selected the
same number? Express your
answer as a common fraction.
Answer:
1
44
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)
1
4
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