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MATHCOUNTS
2006 State Competition
Countdown Round

1. Bobby stands 10 feet from Sam,
and Eric stands 8 feet from Bobby.
What is the least number of feet
that Eric could be from Sam?
Answer: 243
2. Two numbers have a
difference of 3 and a sum of
31. What is the larger of the
two numbers?
Answer: 1320 kilometers
• 3. Arthur walks six blocks east and
then 12 blocks north. If each block is
one-third of a mile, how many miles
did he walk?
Answer: 3.5 (quarts)
• 4. Semicircles of diameter 2” are
lined up as shown. What is the
area, in square inches, of the
shaded region in a 1-foot length
of this pattern? Express your
answer in terms of π.
Answer: 1/6
• 5. If x − 2x + 3x = 100, what is the
value of x?
Answer: 400
• 6. A computer program
reduces the length of each side
of an 8” by 8” square
photograph by 25%. What is
the area, in square inches, of
the reduced photo?
Answer: 30 (students)
• 7. In a set of four consecutive
positive integers, the smallest
integer is less than half the
largest integer. What is the
largest integer that could be in
the set?
Answer: 55 (miles per hour)
• 8. Using the cost structure shown
below, a customer must use more
than x minutes per month for
Tellfon to be the better deal. What
is the value of x?
• Company Monthly Charge
Cost per Minute of Use
• Tellfon
$6 $0.05
• Celltell
$0 $0.10
Answer: 18 (sq centimeters)
• 9. What is the value of (43 + 23) −
(33 + 13)?
Answer: 5 (9s)
• 10. The 24th day of a particular
month is a Saturday. On what day
of the week was the first day of
that month?
Answer: 400 (cartons)
• 11. Five friends went to a
restaurant and agreed to share
the bill equally. However, one
person forgot his wallet so the
other four friends’ portions of the
bill went up by $4 each. How
many dollars was the total bill?
Answer: 20 (percent)
• 12. What common fraction is
equal to half of 80%?
Answer: 6 (integers)
• 13. What is the value of 99 × 99?
Answer: 21 (units)
• 14. Abbie, Ben and Cali will sit in
a row in a theater. What is the
probability that they will sit in
alphabetical order, from left to
right? Express your answer as a
common fraction
Answer: 90 (cubic inches)
• 15. Starting on June 1, on each
odd-numbered date Kirsten earns
$1 and on each even-numbered
date she earns $2. On what
month and date will her total
earnings first reach or exceed
$50?
Answer: 2012
• 16. What is the product of the
reciprocals of 5/4 and10/3,
expressed as a percent?
Answer: 2/3
• 17. If a Y b is defined as a Y b =
a2 − 2ab + b2, what is the value
of 3 Y 2?
Answer: 2.5 dollars or 2
dollars, 50 cents
• 18. If 10% of the people polled
who dislike radio also dislike
music, and 30% of the 1200
people polled disliked radio, how
many people polled disliked both?
Answer: 12 (sequences)
• 19. Analise’s resting heart
rate is 75 beats per minute.
When she sleeps, it slows to
60 beats per minute. What
percent decrease is this?
Answer: 36 (triangles)
• 20. The surface area of a
particular cube is 600 square
inches. When the edges of the
cube are doubled in length, what
is the volume of the new cube, in
cubic inches?
Answer: 4
33. A bottling plant fills 0.5-liter
bottles from a 56-liter tank. A flat
consists of 24 bottles. How many
complete flats can be filled from a
single tank?
Answer: 4 (flats)
34. What is the sum of all the prime
numbers between 1 and 10?
Answer: 17
35. Express 1/(1+1/(1+3/4))
as a common fraction.
Answer: 7/11
36. A tree grew from a height of 6 feet
to 15 feet. By what percent did its
height increase?
Answer: 150 (percent)
37. Suppose that 50! is written in the
form N × 10x, where N is an integer.
What is the largest possible value of
x?
Answer: 12
38. A ball is dropped from 10 feet high
and always bounces back up half the
distance it just fell. After how many
bounces will the ball first reach a
maximum height less than 1 foot?
Answer: 4 (bounces)
39. For the three identical squares
shown, points A, B and C are
vertices, and AB = 2 5 cm. What is
the length of AC, in centimeters?
Express your answer in simplest
radical form.
A
B
C
Answer: 2 10
40. What is the value of
(x + 1 – x)! ÷ (x – x + 1)! in simplest
form??
Answer: 1
41. What two-digit positive integer is
one more than a multiple of 2, 3, 4, 5
and 6?
Answer: 61
42. In this figure, each pair of
adjacent sides is perpendicular.
The figure is to be divided into
12 equal pieces having the same
size and shape. The area of one
of these pieces can be expressed
in the form axⁿ. What is the
product of a and n, expressed as
a common fraction?
2X
X
X
X
Answer: 1/2
43. What is the value of 1/8 of 20% of
1/2 of 480?
Answer: 6
44. What is the value of 3! ⋅3!
expressed as a positive integer?
Answer: 6
45. What is the sum of the final three digits
of the integer representation of 5100?
Answer: 13
46. A cell phone plan costs $50 per
calendar month. At this rate, how many
dollars does it cost for 10 years?
Answer: 6000 (dollars)
47. How many positive integers less than
200 are divisible by 2, 3 and 5?
Answer: 6 (integers)
48. A patient takes one pill every 8 hours
until the whole bottle of 30 pills is gone.
How many hours elapsed between the
time the first pill was taken and the time
the last pill was taken?
Answer: 232 (hours)
49. The cost of five pencils and one pen is
$2.50, and the cost of one pencil and two
pens is $1.85. What is the cost of two
pencils and one pen?
Answer: 1.45 dollars
50. January 1, 2007 was a Monday. In
2007, how many Fridays are there?
Answer: 52 (Fridays)
51. What is the total surface area, in square
inches, of a closed, rectangular box 4
inches wide, 6 inches long and 3 inches
tall?
Answer: 108 (sq inches)
52. A class of 40 children is 60% boys.
How many boys would have to be
replaced by girls to make a majority of
the class girls?
Answer: 5 (boys)
53. If f (x) = |x + 3| and g(x) = x2 − 6 ,
what is the value of f (g(2))?
Answer: 1
54. A pitcher contains 4 1/2 cups of water.
A 1/3 -cup scoop is used to transfer water
from the pitcher to a bowl. How many
scoops are needed until the bowl has
more water than the pitcher?
Answer: 7 (scoops)
55. Grandma gave Bryce and Carter some
raisins. Bryce received 6 more raisins
than Carter, and Carter received half the
number of raisins Bryce received. How
many raisins did Bryce receive?
Answer: 12 (raisins)
56. Kyle has 112 unit cubes. If no unit
cube may be cut into smaller pieces,
what is the minimum number of
additional unit cubes he needs in order to
build a solid cube using all the unit
cubes?
Answer: 13 (cubes)
57. When a positive number is multiplied
by the sum of twice the number and half
the number, the result is the original
number. What is the number? Express
your answer as a common fraction.
Answer: 2/5
58. Given that 0.5 liters is approximately
1.05 pints, how many pints are in one
liter? Express your answer as a decimal
to the nearest tenth.
Answer: 2.1 (pints)
59. How many common fractions with a
value between 0 and 1 have 6 as their
denominator?
Answer: 2 (fractions)
60. The product of three different positive
integers is equal to 7³. What is the sum of
the three integers?
Answer: 57
61. How many inches are in 2 1/6 yards?
Answer: 78 (inches)
62. A ferry trip departs from the ferry dock
every 15 minutes. Yesterday the first
ferry trip left at 8 a.m. and the last ferry
trip left at 5 p.m. How many ferry trips
left from the ferry dock yesterday?
Answer: 37 (trips)
63. The ones digit of the product of four
consecutive positive integers is 4. If this
product is greater than 1000, what is the
sum of the four smallest such integers?
Answer: 30
64. A rectangular box has a volume of 108
cubic feet. How many cubic yards is the
volume of this box?
Answer: 4 (cubic yards)
65. What is the positive difference between
the greatest and the least member of the
set { 3/7, 4/3, 11/8, 6/16 }? Express your
answer in simplest form.
Answer: 1
66. A 2 inch by 2 inch square has its area
increased by 21 square inches producing
a new square. How many inches long is a
side of the new square?
Answer: 5 (inches)
67. The radius of a circle is decreased by
10%. By what percent will the area be
decreased?
Answer: 19 (percent)
69. What is the ratio of the volume of a
cube with edge length six inches to the
volume of a cube with edge length one
foot? Express your answer as a common
fraction.
Answer: 1/8
70. Anne donates 4% of her total income
each year to charity. If her 2005
donations totaled $1,308, what was her
total 2005 income, in dollars?
Answer: 32,700 (dollars)
71. An exhibit hall in a convention center
measures 20 by 30 yards. A square area
of 9 square yards must be reserved in one
corner for registration. How many square
yards will remain for exhibit space?
Answer: 591 (sq yards)
72. Carla spent two-fifths of her savings on
a bracelet and had $9 left over. How
many dollars did Carla’s bracelet cost?
Answer: 6 (dollars)
73. If the angles of a trapezoid form an
arithmetic sequence and the smallest
angle measures 60°, what is the measure
of the largest angle, in degrees?
Answer: 120 (degrees)
74. What is the length, in units, of the
shortest possible diagonal of a rectangle
with all integer side lengths and
perimeter 26 units? Express your answer
in simplest radical form.
Answer:
85
75. When the value of y is doubled and
then this increased value is divided by 5,
the result is 10. What is the value of y?
Answer: 25
76. If x² - 2x = 0 and x ≠ 0, what is the
value of x?
Answer: 2
77. For what value of x is the expression2
(2x3 +3)/ (x2-20x+100) not defined?
Answer: 10
78. What is the largest prime factor of (302
- 72)?
Answer: 37
79. How many cubic inches are in twothirds of one cubic foot?
Answer: 1152 (cu inches)
80. The side of a square is increased by
50%. By what percent is the area
increased?
Answer: 125 (percent)