MATHCOUNTS
2008 Chapter Competition
Countdown Round

1. The forecasted
maximum wind speeds,
in mph, for the next five
days are 13, 6, 10, 5 and
8. What is the range of
the wind speeds, in miles
per hour?
2. In the geometric sequence
with a first term of 6 and a
second term of -6, what is the
205th term?
3. A segment with endpoints at
A(2, -2) and B(14, 4) is
extended through B to point C.
If BC = ( )AB, what are the
coordinates for point C?
Express your answer as an
ordered pair.
4. If a snack-size tin of peaches
has 40 calories and is 2% of a
person’s daily caloric requirement,
how many calories fulfill a
person’s daily caloric
requirement?
5. If Ellen chooses a day of
the week at random, what is the
probability that its English
spelling uses the letter “n”?
Express your answer as a
common fraction.
6. A catalogue sells pencils
two ways: (1) 50 cents each
and (2) $2 per pack of 12.
What is the least expensive
purchase price for exactly
30 pencils?
7. The five highest kick return
averages, in yards, in the NFL’s
history were 25.0, 24.3, 24.1,
23.8 and 23.4. What is the median
of these five averages? Express
your answer as a decimal to the
nearest tenth.
8. How many 4-inch by 4-inch
squares can be cut from a
rectangular piece of leather
measuring 4 feet by two-thirds
of a yard?
9. The product of three
consecutive integers is
720. What is the largest
of these integers?
10. Margo walks to her friend’s
house in 10 minutes. Using the
same route, it takes Margo 20
minutes to get back home. If her
average walking rate for the entire
trip is 4 miles per hour, how many
total miles did she walk?
11. Adjacent sides of Figure 1 are
perpendicular. Four sides of Figure 1 are
removed to form Figure 2. What is the total
length, in units, of the segments in Figure 2?
1
1
8
2
6
Figure 1
Figure 2
12. What is the smallest possible
number of whole 2-by-3 nonoverlapping rectangles needed to
cover a square region exactly,
without extra over-hangs and
without gaps?
13. What is the area, in square
units, of the four triangular faces
of a right, square-based pyramid
that has base edges measuring 6
units and lateral edges measuring
5 units?
14. For how many integer
2
values of x is x < 7x?
15. What is the greatest
possible area, in square units,
of a rectangle with a
perimeter of 16 units?
16. A 5×5×5 cube is
formed by assembling 125
unit cubes. Nine unit
squares are painted on each
of the six faces of the cube
according to the pattern
shown. How many of the
125 unit cubes have no
paint on them?
17. What is the greatest common
divisor of 315 and 108?
18. What is 20% of 50% of 80?
19. A rocket traveled 12,000 miles
in the first four hours. Then the
rocket traveled 8000 miles in the
next four hours. What was the
rocket’s average speed, in miles
per hour, for these 8 hours?
20. If Kelly gets to keep an eighth
of the sales of the store, how
many dollars must the sales of the
store be in order for Kelly to get
$54?
21. Xanthia can read 100 pages per
hour and Molly can read 50 pages per
hour. If they each read the same
book, and the book has 225 pages,
how many more minutes than
Xanthia would it take for Molly to
finish reading the book?
22. The first four triangular numbers
are 1, 3, 6 and 10 where each
successive number can be found by
adding a row of dots with one more
dot than the previous row. What is
the sixth triangular number?
1
3
6
10
23. John has 5 quarters and David has
5 dimes. John owes David 35 cents.
John should give David m quarters
and John should get n dimes from
David so that John has paid his debt
exactly. What is the value of m + n?
24. If f(x) = 2 for all real
numbers x, what is the value
of f(x + 2)?
25. Increasing a number by 10% is
the same as multiplying it by y.
What is the value of y? Express
your answer as a decimal to the
nearest tenth.
26. Betty draws one disc from a jar
that contains 2 red discs, 5 green
discs and 8 yellow discs. What is the
probability that her selection is not a
red disc? Express your answer as a
common fraction.
27. A triangle has side lengths
of 8, 15 and 17 units. What is
the area of the triangle, in
square units?
28. Jack bought an item for
$300. It had been discounted
by 20%. How many dollars
was the discount?
29. At what point does the line
containing the points (1, 7) and
(3, 11) intersect the y-axis?
Express your answer as an
ordered pair.
30. Quadrilateral ABCD is a square with
area 16 square inches. The figure
represents the pieces of a Chinese tangram
in which all the triangles are isosceles and
piece “e” is a square. What is the area of
the gray piece, in square inches?
A
B
e
D
C
31. Anna purchased 1 pound of
fusilli pasta for $3.40 and a
12-ounce package of spaghetti
for $2.20. There are 16 ounces
in 1 pound. What was the
average price per ounce, in
cents, for her total purchase?
32. Tim can do one-third of
the job in 40 minutes. At
this rate, how many total
minutes will it take him to
do the complete job?
33. Let P =
What is the value of P if
n = 2007? Express your
answer as a common
fraction.
.
34. An 80-pound box of apples
will have 10% of the apples
spoil. At the same rate, how
many pounds of spoiled apples
will there be for every 2160
pounds of unspoiled apples?
35. The four midpoints of adjacent
sides of a square are joined to
form a smaller square. What is the
ratio of the area of the smaller
square to the area of the original
square? Express your answer as a
common fraction.
36. What is the height of Jack’s
house, in feet, if the house casts
a shadow 56 feet long at the
same time a 21-foot tree casts a
shadow that is 24 feet long?
Express your answer to the
nearest whole number.
37. What is the sum of the
solutions of the equation
x(2x - 9) = 0? Express your
answer as a common fraction.
38. A recent study found that 60% of
men and 80% of women surveyed
support increased funding for
particular medical research. The
study surveyed 100 men and
900 women. What was the overall
percent of the people surveyed who
supported increased funding?
39. What is the value of x
which satisfies
?
40. A bag of grapes is to be distributed
evenly to 5 kids in a class, and the
grapes that are left over will be
thrown out. If each student receives
the greatest possible number of
grapes, what is the greatest possible
number of grapes that could be
thrown out?
41. The operation ◊ is defined
by a ◊ b = ab2 - b + 1. What is
the value of (-1) ◊ 6?
42.What is the smallest possible
area, in square units, of a triangle
whose side lengths are all
integers? Express your answer as
a common fraction in simplest
radical form.
43. When (0.2)4 is written as a
decimal, what is the product
of the last two digits?
44. A dose of a shrinking potion
makes a person 50% smaller than
they were before they drank it, and
eating a growth cake makes a person
50% larger than they were before
they ate it. After drinking 2 doses of
shrinking potion and eating 1 growth
cake, what is the ratio of Alice’s
final size to her original size?
Express your answer as a common
fraction.
45. If you continue this pattern in
which each line-segment extremity is
replaced by a gradually smaller Y in
the next figure, in the manner shown,
how many endpoints will Figure 5
have?
Figure 1
Figure 2
Figure 3
46. A right triangle has leg lengths
of 8 and 15 units. What is the
length, in units, of the median to
the hypotenuse? Express your
answer as a common fraction.
47. An 8.5-by-11-inch piece of paper
is folded in half repeatedly (never
being unfolded), each time
shortening the then longer side. What
is the length of the longest side, in
inches, immediately after the second
fold? Express your answer as a
decimal to the nearest tenth.
48. What is half of the absolute
value of the difference of the
squares of 18 and 16?
49. A right triangle has an area
of 120 square units, and a leg
length of 24 units. What is
the perimeter of the triangle,
in units?
50. The number 1341 can be
written as the sum of three
consecutive positive integers.
What is the largest of these
integers?
51. If the consecutive integers
from 50 to 1 were written as
5049484746..., what would
be the 67th digit to be
written?
52. A square with side length 5 inches
and a 3-by-6-inch rectangle are each
drawn on the coordinate plane such
that there are 12 square inches of
overlap. What is the area, in square
inches, of the portion of the square
that is not overlapping the rectangle?
53. One cookie and two
brownies cost the same as
two cookies and one
brownie. One cookie and
one brownie together cost
$1. What is the cost, in
cents, of one cookie?
54. An isosceles, obtuse triangle has
one angle with a degree measure that
is 50% larger than the measure of a
right angle. What is the measure, in
degrees, of one of the two smallest
angles in the triangle? Express your
answer as a decimal to the nearest
tenth.
55. Consecutive powers of 3 are
added to form this sequence:
30, 30 + 31, 30 + 31 + 32, and so
on. What is the simplified
value of the fourth term of the
sequence?
56. Five years ago, a computer that
worked half as fast as a current
computer cost twice as much
money. The current computer’s
speed to price ratio is what
percent of the older computer’s
speed to price ratio?
57. How many integers belong
to the arithmetic sequence 13,
20, 27, 34, ..., 2008?
58. A rectangular garden shares
one side with one side of a
house. An adjacent side of the
garden is eight feet long. If the
area of the garden is 184 square
feet, what is the length, in feet,
of the common side?
59. The equation x2 + ax = -14
has only integer solutions for
x. If a is a positive integer,
what is the greatest possible
value of a?
60. A right triangle has leg
lengths of 7 and 24 units.
A second triangle is similar
to the first triangle and has a
hypotenuse of 100 units.
What is the length, in units,
of the shorter leg of the
second triangle?
61. There is a total of 20 quarters
stacked in four piles. The first
pile has 3 fewer than the second
pile. The second pile has 2 more
than the third pile. The fourth
pile has twice as many as the
second pile. How many quarters
are in the fourth pile?
62. A cube with an edge length
of 4 units has the same
volume as a square-based
pyramid with base edge
lengths of 8 units and a
height of h units. What is the
value of h?
63. Bill pays $49 for an item
after it has been discounted
30%. What was the original
price of the item?
x
4
x+y
5
64. If = 16 and
= 625,
what is the value of y?
65. Simone has a collection of
coins consisting of 1 quarter,
1 nickel and 1 penny. Using at
least two coins from the
collection, how many different
sums of money are possible?
66. What is the value of 625
times 44?
67. In a music collection, there are
100 songs with an average length
of exactly 180 seconds. One song
is added and the new average
length is exactly 181 seconds.
How long, in seconds, was the
added song?
68. What is the
sum, in degrees,
of the measures
of the interior
angles in the
convex hexagon
shown?
69. Dave rode 30 miles at
10 miles per hour and 10
miles at 30 miles per hour.
What was his average speed,
in miles per hour, for the
entire ride?
70. If
and
, what is
the smallest positive integer
n such that x < y?
71. If a:b = 1:3 and b is a
positive, one-digit integer,
what is the greatest possible
value of a?
72. Apartment rentals in Fairview
run approximately $0.90 per
square foot. Jillian has
determined that she can afford
$630 per month for rent. What is
the largest apartment, in square
feet, she should consider at the
given rate?
73. The operation ☼ is defined
3
by x☼y = x +y. What is the
value of 2☼4?
2
y
74. If = 36, what is the
3
greatest value of y ?
75. The digits 1, 3 and 5 are
each used once to form each
of the possible three-digit
positive integers. The threedigit integers are listed from
greatest to least. Which
integer is listed fifth?
76. Parallelogram ABCD has
A(0, 1), B(2, 1) and C(4, 0)
as three of its vertices. What
is the sum of the coordinates
of point D?
77. If
, what is
the value of
?
Express your answer as a
decimal to the nearest tenth.
78. What is the value of n in
the equation n + (n + 1) +
(n + 2) = 9?
79. Charlie rides her bike on a
route that is 22 miles long. It
takes her 165 minutes to
complete the ride. What is
her resulting average speed,
in miles per hour?
80. A regular polygon has
sides of length 5 units and an
exterior angle of 120 degrees.
What is the perimeter of the
polygon, in units?