1. Compute the product
4 6 8 10 12 14
⋅ ⋅ ⋅ ⋅ ⋅ .
3 4 5 6 7 8
2. What is the least integer greater than
1. ______________
300 ?
2. ______________
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.
3. ______________
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?
4. ______________
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?
5. ______________
6. What is the remainder when 2003 is divided by 11?
6. ______________
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?
7. ______________
8. The sum of the first and third of three consecutive integers is
118. What is the value of the second integer?
8. ______________
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?
9. ______________
10. If 2ab = 12, evaluate 8a2b2.
10. ______________
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?
11. ______________
12. Each corner of a cube is cut off to leave an
equilateral triangle as shown. How many faces does
the new polyhedron have?
12. ______________
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?
13. ______________
2
4
14. What is the eighth term in the arithmetic sequence , 1 , , ...?
3
3
Express your answer is simplest form.
14. ______________
©2003 MATHCOUNTS Foundation: 2003 State Countdown Round
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.
15. ______________
16. What is the greatest integer value of n such that 3n < 10,000 ?
16. ______________
17. What is the units digit of the product of the first 100 prime
numbers?
17. ______________
18. Solve for m: (m − 4)
1
= 8
−1
.
18. ______________
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?
19. ______________
20. Which integer is closest to the cube root of 100?
20. ______________
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?
21. ______________
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 π .
22. ______________
23. Mary completes a 15-mile race in 2.5 hours. What is her
average speed in miles per hour?
23. ______________
24. If 15% of N is 45% of 2003, what is the value of N?
24. ______________
25. How many digits are in the whole-number representation of
10100 – 9100?
25. ______________
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?
26. ______________
27. How many square units are in the area of the largest square that
can be inscribed in a circle with radius 1 unit?
27. ______________
28. Express the sum as a common fraction:
.1 + .02 + .003 + .0004 + .00005.
28. ______________
3
10
8
29. Compute: 2 7 − 26 . Express your answer in simplest form.
2 −2
29. ______________
©2003 MATHCOUNTS Foundation: 2003 State Countdown Round
30. What is the 2003rd term of the sequence of odd numbers 1, 3,
5, 7, ... ?
30. ______________
31. For how many different digits n is the two-digit number “6n”
divisible by n?
31. ______________
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?
32. ______________
33. Calvin is 150 cm tall, which is 75% of Darryl’s height. How
many centimeters tall is Darryl?
33. ______________
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?
34. ______________
35. For what integer x is
3 x 7
< < ?
5 7 9
35. ______________
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?
36. ______________
8 ⋅18 ⋅ 20 ⋅ 5 .
37. ______________
38. What percent of 16 is 50% of 50% of 48?
38. ______________
39. Solve for n: 2n ⋅ 4n = 64n − 36 .
39. ______________
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))?
40. ______________
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?
41. ______________
42. The square with vertices (–1, –1), (1, –1), (–1, 1) and (1, 1) is
cut by the line y = 2x + 1 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.
42. ______________
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?
43. ______________
37. Compute:
©2003 MATHCOUNTS Foundation: 2003 State Countdown Round
44. A digital clock reads 8:30. What will it read in 100 minutes?
44. ______________
45. How many integers n satisfy (n − 2)(n + 4) < 0 ?
45. ______________
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?
46. ______________
47. How many three-digit positive integers exist, all of whose
digits are 2’s and/or 5’s?
47. ______________
48. Define #N by the formula #N = .5(N) + 1 . Calculate
#(#(#50)).
48. ______________
49. What is the 100th term of the arithmetic sequence 6, 10, 14,
18, ...?
49. ______________
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 the smaller polyhedron is what
common fraction of the volume of the whole cube?
50. ______________
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.
51. ______________
52. What is the median of the first ten positive integers? Express
your answer as a decimal to the nearest tenth.
52. ______________
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.
53. ______________
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.
54. ______________
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.
55. ______________
©2003 MATHCOUNTS Foundation: 2003 State Countdown Round
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.
56. ______________
57. A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does
a regular octagon have?
57. ______________
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?
58. ______________
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.
59. ______________
60. What is the value of the number which lies 23 of the distance
from 98.36 to 100.25 on a number line? Express your answer
as a decimal to the nearest hundredth.
60. ______________
61. What is the units digit of 2323?
61. ______________
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.
62. ______________
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?
63. ______________
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?
64. ______________
65. What is the sum of all positive two-digit odd integers?
65. ______________
66. Two numbers are in the ratio of 5:6. Their sum is 77. What is
the positive difference between the two numbers?
66. ______________
67. Each successive term in the sequence 2048, 512, 128, x,
y, 2, 21 , 81 , ... is obtained by multiplying the previous term by a
constant. What is the value of x + y?
67. ______________
©2003 MATHCOUNTS Foundation: 2003 State Countdown Round
68. Karen needs pieces of lace each 23 yards long. How many
whole pieces can she cut from a 9-yard piece of lace?
68. ______________
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?
69. ______________
70. Which of the following is the highest annual wage: $600 per
week, $2500 per month or $31,000 per year?
70. ______________
71. What is the remainder when the sum 12 + 22 + 32 + ... + 102 is
divided by 11?
71. ______________
72. What is the greatest integer value of n such that 25! is divisible
by 10n?
72. ______________
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?
73. ______________
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?
74. ______________
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?
75. ______________
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 ?
76. ______________
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?
77. ______________
78. If f (x) = 3x2 – 5, what is the value of f ( f (–1)) ?
78. ______________
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.
79. ______________
80. Simplify: 3!( 2 3 + 9 ) ÷ 2 .
80. ______________
©2003 MATHCOUNTS Foundation: 2003 State Countdown Round