Download CHAPTER Countdown 2013 - FINAL

2013 Chapter Competition
Countdown Round
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Sample Question
A 3-ounce can of tomato sauce
costs $1.68. In cents, what is the
price per ounce?
Sample Question
Answer: 56 (cents)
2013 Chapter Competition
Countdown Round
1. How many CDs can Edward
buy for $14 each and spend the
same amount he would spend to
buy 60 CDs for $7 each?
Answer: 30 (CDs)
2. What is the smallest prime that
is a factor of the sum 5! + 1?
Answer: 11
3. What is the value of x if
3x + 7 = 22?
Answer: 5
4. The sum of the first n odd
positive integers is 64. What is
the value of n?
Answer: 8
5. A bookstore employee is
given a 20% discount off the
retail price of any book.
Assuming there is no tax, how
many dollars does she pay for a
book with a retail price of $55?
Answer: 44 (dollars)
6. If an isosceles triangle has
base angles that are each twice
the measure of the smaller angle,
what is the degree measure of
one of the base angles?
Answer: 72 (degrees)
7. Tim’s math homework is on
five consecutive pages in the
math textbook, and the sum of
those page numbers is 630.
What is the page number of the
next page after these five
homework pages?
Answer: (page) 129
8. The intersection of a circular
region of radius 3 inches and a
circular region of radius 2 inches
has area π in2. In square inches,
what is the area of the total region
covered by the two circular
regions? Express your answer in
terms of π.
Answer: 12π
2
(in )
9. What is the value of the
product 47 × 53?
Answer: 2491
10. What is the ratio of the
number of degrees in the interior
angle of a regular pentagon to
the number of degrees in the
interior angle of a regular
hexagon? Express your answer
as a common fraction.
9
Answer: 10
11. Marielle buys an equal
number of 50¢ and 75¢ candy
bars and spends $10, not
including tax. How many candy
bars did she buy altogether?
Answer: 16 (candy bars)
12. A restaurant automatically
adds an 18% tip to the bill. If
the tip was $9, what was the
bill before the tip was added, in
dollars?
Answer: 50 (dollars)
13. The integer 12,345 can be
expressed as the sum of two
prime numbers in exactly one
way. What is the larger of the
two primes in this sum?
Answer: 12,343
14. Let x be an integer that
satisfies x4 + 24 = 105. What is
the value of x2 − 24?
Answer: −15
15. Triangle ABC has sides of
length 30, 40 and 50 units. What
is the mean, in degrees, of the
measures of the three angles?
Answer: 60 (degrees)
16. The number 18 can be
written as the sum of nine
consecutive integers. What is
the product of these integers?
Answer: 0
17. When expressed as a common
fraction, what is the value of
2  4  6  8   1340  1342 ?
3  6  9  12   2010  2013
2
Answer: 3
18. What is the probability that
a randomly selected two-digit
positive integer will be a
multiple of 11? Express your
answer as a common fraction.
1
Answer: 10
19. What is the greatest
prime factor of the
difference 642 − 612?
Answer: 5
20. The three circles in this figure are
all tangent. Their centers are collinear.
2
The diameter of the smallest circle is
5
that of the largest circle. What fraction
of the largest circle is gray?
Express your answer as
a common fraction.
12
Answer:
25
21. What is the greatest possible
c
b
value of a where a, b and c are
distinct positive integers less
than 4?
Answer: 9
22. If the five points marked
by dots on the number line
below are equally spaced, what
is the value of y?
3
x
y
2x
Answer: 6
23. What is the number of
square meters in the area of a
square if the length of a
diagonal is 14 meters?
Answer: 98 (m2)
24. How many miles are
traveled driving at 30 mi/h for
30 minutes?
Answer: 15 (miles)
25. What is the median of the
set of all positive factors of 100?
Answer: 10
26. What is the value of
41
9 3
the quotient 4 1 ?

27 9
Answer: 3
27. If a2 − b2 = 10 and a − b = 2,
what is the value of a + b?
Answer: 5
28. When the positive integer x
is divided by each of 4, 5 and 6,
it has a remainder of 3. What is
the sum of the three smallest
possible values of x?
Answer: 189
29. The sum of the interior
angles of a convex polygon
is 900°. How many sides
does the polygon have?
Answer: 7 (sides)
30. How many positive integers
between 100 and 200 are
divisible by 14?
Answer: 7 (integers)
31. What is the least value of
n for which a regular n-gon
has at least 17 diagonals?
Answer: 8
32. What is the value of the
2
2
difference 2018 − 2013 ?
Answer: 20,155
33. What is the denominator
11111
when 2 3 4 5 6 is
written as a common fraction?
Answer: 20
34. What is the range of the set
{121, 142, 163, 184, 106}?
Answer: 78
35. What is the value of the quotient
99
1
2
3
1
1 1 1
100
2
3
4
?
1
101!
Answer: 101
36. John is 3 years older than
Peter. Ten years ago the sum of
their ages was 59. In years, how
old is John now?
Answer: 41 (years old)
37. The legs of a right triangle
3
4
measure 5 inches and 5 inches.
In inches, what is the length of
the hypotenuse?
Answer: 1 (inch)
38. What is the value of
2013  2  2013  2012  2012 ?
2
2
Answer: 1
39. Together three puppies and
three adult dogs weigh 60 pounds.
Two adult dogs and four puppies
have a combined weight of 50
pounds. If each of the puppies has
the same weight and each of the
adult dogs has the same weight,
how many pounds does one puppy
weigh?
Answer: 5 (pounds)
40. How many distinct factors
of 25! are prime numbers?
Answer: 9 (factors)
41. A regular hexagon of side
length 4 inches is divided into four
triangles by three nonintersecting
diagonals. What is the average
area, in square inches, of each of
the four triangles? Express your
answer in simplest radical form.
Answer: 6 3 (in2)
42. What is the value of x if
3
12
x
=
?
Express
your
5
25
answer as a common fraction.
4
Answer: 5
43. Sharon drives 30 mi at 60 mi/h
and 60 mi at 30 mi/h. What is
Sharon’s average speed, in miles
per hour, for the whole trip?
Answer: 36 (mi/h)
44. The three angle measures of a
x x
x
triangle are 2 , 3 and 6 degrees.
What is the degree measure of
the smallest angle?
Answer: 30 (degrees)
45. What is the median of the set
consisting of the first ten prime
numbers?
Answer: 12
46. The sum of the squares of
four consecutive positive
integers is 366. What is the
largest of the consecutive
integers?
Answer: 11
47. The perimeter of a rhombus
is 156 cm. The shorter diagonal
measures 30 cm. What is the
length, in centimeters, of the
longer diagonal?
Answer: 72 (cm)
48. What is the smallest positive
1 a2
integer a such that 2
is an
3
integer?
Answer: 6
49. In the game Yahtzee®, five
standard dice are tossed. What
is the probability that all five
dice show even numbers or all
five show odd numbers?
Express your answer as a
common fraction.
1
Answer: 16
50. What is the value of
0.1 + 0.2 + 0.3 + 0.4 ?
Express your answer as a
common fraction.
10
Answer: 9
51. Five couples went to the
movies together. They all sit in
ten adjacent seats in the same
row. How many different ways
can they be seated if each
couple sits together?
Answer: 3840 (ways)
52. What is the value of
8 ÷ 8 − 8 + 8 × 8?
Answer: 57
53. What is the product of the
least common multiple and the
greatest common factor of 84
and 105?
Answer: 8820
54. Kevin has twice as many
cookies as Aidan and half as
many cookies as Beth. If Aidan
and Beth have 35 cookies
together, how many cookies does
Kevin have?
Answer: 14 (cookies)
55. A round birthday cake was
divided into 16 congruent slices,
3
and of the cake was eaten. The
8
next day, five slices were eaten.
How much of the original cake is
left? Express your answer as a
common fraction.
5
Answer: 16
56. What is the least of three
consecutive integers whose
product is 1716?
Answer: 11
57. Each of the 2898 students at
Descartes Middle School voted for
his or her favorite meal chosen
from a list of six different meals.
Pia created a circle graph to
represent the data. If a 60-degree
sector of the circle graph represents
votes for pizza, how many students
voted for pizza?
Answer: 483 (students)
58. The cost of daily school
lunch increased from $1.80 to
$2.25. What was the percent
increase?
Answer: 25 (percent)
59. When expressed as an
integer, what is the units
digit of 2013!?
Answer: 0
60. Sun-Li sold candy bars for
the glee club for $0.50 each. If
3
she sold of the candy bars that
4
she had and has 12 candy bars
left, how many dollars has she
collected so far?
Answer: 18 (dollars)
61. For what value of n is the
sum of the first n positive
integers equal to 190?
Answer: 19
62. An isosceles triangle with
sides of integer length has a
perimeter of 24 inches. If the
ratio of two of its sides is 2:3,
what is the number of inches in
the length of one of the legs?
Answer: 9 (inches)
63. If g = 4, what is the value
4g  g
of
?
4
Answer: 5
64. A conical pool takes 2 hours
to be filled at a uniform rate to a
depth of 6 ft. How many
minutes does it take to fill it to a
depth of 3 ft?
Answer: 15 (minutes)
65. For how many integers x is
(x − 3)(x + 4) < 0?
Answer: 6 (integers)
66. How many diagonals can be
drawn in a convex dodecagon?
Answer: 54 (diagonals)
67. Eduardo is writing math
problems. He writes 1 problem
on day 1, 2 problems on day 2, 3
problems on day 3, and so on. If
this pattern continues, on what
day does he write the 50th
problem?
Answer: (day) 10
68. What is the 100th term of
the arithmetic sequence
3, 11, 19, 27, …?
Answer: 795
69. Richard is thinking of three
distinct, positive integers. He
tells Barbara their sum is 9, and
he tells Lori that their product is
24. What is the median of
Richard’s three numbers?
Answer: 3
70. Two sides of a triangle
measure 9 units and 11 units. In
units, what is the positive
difference between the measures
of the smallest and the largest
possible integral lengths of the
third side of the triangle?
Answer: 16 (units)
71. What is the sum of all
positive integers from 1 to 30,
inclusive, that are neither
multiples of 2 nor perfect
squares?
Answer: 190
72. In miles, how far will Jill
1
travel if she drives for 3 hours
2
at an average speed of 44 mi/h?
Answer: 154 (miles)
73. What is the mean of the
first five triangular numbers?
Answer: 7
74. Molly has eight U.S. coins
with a total value of 78 cents.
She does not have any halfdollars. How many dimes does
Molly have?
Answer: 2 (dimes)
75. The domain of a function
f(x) is all real numbers and the
range of f(x) is all real numbers
from −1 to 12, inclusive. What
is the maximum value of g(x) if
g(x) = 2f(x – 1) + 3?
Answer: 27
76. In an arithmetic progression
the first term is 0 and the fifth
term is 5. What is the third term?
Express your answer as a
common fraction.
5
Answer:
2
77. The positive real numbers w,
x and y satisfy the equation
2x
= 16yw. If y is tripled and w
5
is halved, by what percent must
x be increased so that the new
values of w, x and y also satisfy
the equation?
Answer: 50 (percent)
78. What is the value of
6.9 × 1015 ÷ (2.3 × 107) × 7.0 × 10−4?
Express your answer in scientific
notation.
Answer: 2.1 × 105
79. How many sides does a
regular polygon have if the
measure of an interior angle is
150 degrees?
Answer: 12 (sides)
80. If g(x) = 2x – 8 and
f(x) = 3x2 + 17x, what is
f(g(3))?
Answer: −22