Transcript 1. What is the sum of the number of faces, vertices and

MATHCOUNTS

2003 State Competition
Countdown Round
1. Compute the product
4 6 8 10 12 14
    
3 4 5 6 7 8
Answer: 16
2. What is the least integer
greater than 300 ?
Answer: 18
3. A number is randomly
selected from the first 20 positive
integers. What is the probability
that it is divisible by 2, 3 or 4?
Express your answer as a
common fraction.
13
Answer:
20
4. A tree is now 12 feet tall and
grows 18 inches a year. In how
many years will the tree be
36 feet tall?
Answer: 16 (years)
5. Stacey is standing in a field.
She walks 11 meters west,
30 meters north, 4 meters west
and finally 22 meters south.
How many meters is she from her
starting point?
Answer: 17 (meters)
6. What is the remainder when
2003 is divided by 11?
Answer: 1
7. Use each of the digits 3, 4, 6,
8 and 9 exactly once to create the
greatest possible five-digit
multiple of 6. What is that
multiple of 6?
Answer: 98,634
8. The sum of the first and third
of three consecutive integers is
118. What is the value of the
second integer?
Answer: 59
9. A palindrome is a number
that reads the same forward and
backward, such as 1331. What is
the palindrome between 500 and
750 that is divisible by 3 and
whose digits have a product of
200?
Answer: 585
10. If 2ab = 12, evaluate 8a2b2.
Answer: 288
11. A candy bar weighs
4.48 ounces. A person eats half
of the bar with the first bite. If a
person eats half of the amount
remaining with each subsequent
bite, how many bites have been
taken when exactly 0.07 ounces
remain?
Answer: 6 (bites)
12. Each corner of a cube is cut
off to leave an equilateral triangle
as shown. How many faces does
the new polyhedron have?
Answer: 14 (faces)
13. Everyone in a class of
30 students takes math and
history. Seven students received
an A in history, 13 received an A
in math and four received an A in
both courses. How many
students did not receive an A in
either course?
Answer: 14 (students)
14. What is the eighth term in the
2
4
arithmetic sequence , 1, , ... ?
3
3
Express your answer in simplest
form.
Answer: 3
15. The diagonals of a regular
hexagon have two possible
lengths. What is the ratio of the
shorter length to the longer
length? Express your answer as a
common fraction in simplest
radical form.
3
Answer:
2
16. What is the greatest integer
value of n such that 3n < 10,000?
Answer: 8
17. What is the units digit of the
product of the first 100 prime
numbers?
Answer: 0
18. Solve for m: (m  4) 
3
1 1
(8)
.
Answer: 6
19. A regular pentagon is rotated
counterclockwise about its center.
What is the minimum number of
degrees it must be rotated until it
coincides with its original
position?
Answer: 72 (degrees)
20. Which integer is closest to
the cube root of 100?
Answer: 5
21. Dora has three candles,
which will burn for 3, 4 and
5 hours, respectively. If she
wants to keep two candles
burning at all times, what is the
greatest number of hours she can
burn the candles?
Answer: 6 (hours)
22. What is the expression for
the area of the circle which has a
circumference of C? Express
your answer as a rational
expression in terms of C and  .
2
Answer:
C
4
(or equivalent)
23. Mary completes a 15-mile
race in 2.5 hours. What is her
average speed in miles per hour?
Answer: 6 (miles per hour)
24. If 15% of N is 45% of 2003,
what is the value of N?
Answer: 6009
25. How many digits are in the
whole-number representation of
100
100
10 – 9 ?
Answer: 100 (digits)
26. One night the temperature
fell 3 degrees every hour from
6 p.m. through midnight. The
temperature was 17 degrees at
6 p.m. What was the temperature
at midnight that night?
Answer: –1 (degrees)
27. How many square units are
in the area of the largest square
that can be inscribed in a circle
with radius 1 unit?
Answer: 2 (square units)
28. Express the sum as a
common fraction:
.1 + .02 + .003 + .0004 + .00005.
2469
Answer:
20,000
2 2
29. Compute: 7 6 . Express
2 2
10
8
your answer in simplest form.
Answer: 12
rd
2003
30. What is the
term of
the sequence of odd numbers 1,
3, 5, 7, … ?
Answer: 4005
31. For how many different
digits n is the two-digit number
“6n” divisible by n?
Answer: 6 (digits)
32. If 1 skip equals 4 hops, and
1 jump equals 3 skips, what is the
total number of hops in a hop, a
skip and a jump?
Answer: 17 (hops)
33. Calvin is 150 cm tall, which
is 75% of Darryl’s height. How
many centimeters tall is Darryl?
Answer: 200 (centimeters)
34. An edge of a square can be
expressed as 4x – 15 meters or as
20 – 3x meters. What is its area
in square meters?
Answer: 25 (square meters)
35. For what integer x is
3 x 7
 
?
5 7 9
Answer: 5
36. One caterer charges a basic
fee of $100 plus $15 per person.
A second caterer charges a basic
fee of $200 plus $12 per person.
What is the least number of
people for which the second
caterer is cheaper?
Answer: 34 (people)
37. Compute:
8  18  20  5 .
Answer: 120
38. What percent of 16 is 50% of
50% of 48?
Answer: 75 (percent)
39. Solve for n: 2  4  64
n
n
n  36
.
Answer: 72
40. Let GCF (a, b) denote the
greatest common factor of a and
b and let LCM (a, b) denote the
least common multiple of a and
b. What is LCM (GCF (24, 32),
GCF (12, 18))?
Answer: 24
41. The areas of squares A1 and
A2 are 25 square centimeters and
49 square centimeters
respectively. What is the number
of square centimeters in the area
of rectangle A3?
A2
A1
A3
Answer: 35 (square centimeters)
42. The square with vertices (–
1, –1), (1, –1), (–1, 1) and (1,
x
y2
1
1) is cut by the line
into
a triangle and a pentagon. What
is the number of square units in
the area of the pentagon?
Express your answer as a decimal
to the nearest hundredth.
Answer: 3.75 (square units)
43. A horse 24 feet from the
center of a merry-go-round
makes 32 revolutions. In order to
travel the same distance, how
many revolutions would a horse
8 feet from the center have to
make?
Answer: 96 (revolutions)
44. A digital clock reads 8:30.
What will it read in 100 minutes?
Answer: 10:10
45. How many integers n satisfy
(n – 2)(n + 4) < 0?
Answer: 5 (integers)
46. A toy store manager received
a large order of Mr. Slinkums just
in time for the holidays. The
manager places 20% of them on
the shelves, leaving the other
120 Mr. Slinkums in storage.
How many Mr. Slinkums were in
this order?
Answer: 150 (Mr. Slinkums)
47. How many three-digit
positive integers exist, all of
whose digits are 2’s and/or 5’s?
Answer: 8 (integers)
48. Define #N by the formula
#N = .5(N) + 1.
Calculate #(#(#50)).
Answer: 8
49. What is the 100th term of the
arithmetic sequence 6, 10, 14,
18, … ?
Answer: 402
50. On the given cube, M is the
midpoint of segment BC and N is
the midpoint of segment AD . The
plane MNEF slices the solid cube
into two polyhedra. The volume of
E
F
the smaller polyhedron
B
A
is what common
fraction of the volume N
M
of the whole cube?
D
C
1
Answer:
4
51. A baby blue whale weighed
3 tons at birth. When it was
10 days old, it weighed 4 tons. If
its rate of growth was constant
for the first 60 days of its life,
how many tons did this whale
weigh when it was 45 days old?
Express your answer as a decimal
to the nearest tenth.
Answer: 7.5 (tons)
52. What is the median of the
first ten positive integers?
Express your answer as a decimal
to the nearest tenth.
Answer: 5.5
53. The length of each edge of a
cube is decreased by 40%. By
what percent does the volume of
the cube decrease? Express your
answer to the nearest tenth.
Answer: 78.4 (percent)
54. Assuming that the birth of a
boy or a girl is equally likely,
what is the probability that the
three children in a family include
at least one boy and one girl?
Express your answer as a
common fraction.
3
Answer:
4
55. Two different numbers are
randomly selected from the set
{1, 2, 3, 4} and they are
multiplied. What is the
probability that the product is
even? Express your answer as a
common fraction.
5
Answer:
6
56. The cost of a meal at a
restaurant was $15 before tax and
tip. If the 7% tax and an 18% tip
are each based solely upon the
cost of the meal, what is the total
cost in dollars of the meal, tax
and tip? Express your answer as a
decimal to the nearest hundredth.
Answer: 18.75 (dollars)
57. A diagonal of a polygon is a
segment joining two
non-consecutive vertices of the
polygon. How many diagonals
does a regular octagon have?
Answer: 20 (diagonals)
58. Let S be the set of all
three-digit numbers formed by
three consecutive digits in
increasing order. What is the
greatest common factor of all the
three-digit numbers in S?
Answer: 3
59. To take quizzes, each of
30 students in a class is paired
with another student. If the
pairing is done randomly, what is
the probability that Margo is
paired with her best friend, Irma?
Express your answer as a
common fraction.
1
Answer:
29
60. What is the value of the
2
number which lies 3 of the
distance from 98.36 to 100.25 on
a number line? Express your
answer as a decimal to the
nearest hundredth.
Answer: 99.62
61. What is the units digit
23
of 23 ?
Answer: 7
62. In a recent year, the number
of residents of Palm Springs was
45,000 and there were 3 million
visitors to Palm Springs. What
was the average number of
visitors per resident of Palm
Springs? Express your answer to
the nearest whole number.
Answer: 67 (visitors)
63. A couple has the opportunity
to choose three of five concerts
for the coming season. In how
many ways can they choose the
three concerts?
Answer: 10 (ways)
64. If 4 horses eat 4 bales of hay
in 4 days, how many days will it
take 20 horses to eat 30 bales of
hay?
Answer: 6 (days)
65. What is the sum of all
positive two-digit odd integers?
Answer: 2475
66. Two numbers are in the ratio
of 5:6. Their sum is 77. What is
the positive difference between
the two numbers?
Answer: 7
67. Each successive term in the
sequence 2048, 512, 128, x, y, 2,
1 1
is
obtained
by
,
,
...
2 8
multiplying the previous term by
a constant. What is the value of
x + y?
Answer: 40
68. Karen needs pieces of lace
2
each 3 yards long. How many
whole pieces can she cut from a
9-yard piece of lace?
Answer: 13 (pieces)
69. The side lengths of a triangle
are 14 cm, 48 cm and 50 cm.
How many square centimeters
are in the area of the triangle?
Answer: 336 (square
centimeters)
70. Which of the following is the
highest annual wage: $600 per
week, $2500 per month or
$31,000 per year?
Answer: 600 (dollars per week)
71. What is the remainder when
the sum 12 + 22 + 32 + … + 102 is
divided by 11?
Answer: 0
72. What is the greatest integer
value of n such that 25! is
n
divisible by 10 ?
Answer: 6
73. The capacity of a school is
1100 students and the current
enrollment is 980 students. If the
student population increases at a
rate of 5 percent per year, in how
many years will the enrollment
exceed the capacity?
Answer: 3 (years)
74. A rectangle has a perimeter
of 30 units and its dimensions are
whole numbers. What is the
maximum possible area of the
rectangle in square units?
Answer: 56 (square units)
75. A yogurt shop sells four
flavors of yogurt and has six
different toppings. How many
combinations of one flavor and
two different toppings are
available?
Answer: 60 (combinations)
76. What is the units digit of the
sum of the nine terms of the
sequence 1! + 1, 2! + 2,
3! + 3, …, 8! + 8, 9! + 9 ?
Answer: 8
77. Ashley and Chris each
prepare a guest list for a party.
No name appears on both lists.
Half of Chris’ guests attend and a
third of Ashley’s guests attend for
a total of 100 guests. Ashley’s list
contained 180 names. How many
names were on Chris’ list?
Answer: 80 (names)
78. If f (x) =
– 5, what is the
value of f ( f (–1)) ?
2
3x
Answer: 7
79. When rolling two fair,
standard six-faced dice, what is
the probability of rolling the
same number on both dice?
Express your answer as a
common fraction.
1
Answer:
6
80. Simplify: 3!(2  9 )  2 .
3
Answer: 33