MATHCOUNTS

2005
State Competition
Countdown Round
Problems 1–80
This section contains problems to be used in
the Countdown Round.
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Shell Oil Company
Texas Instruments Incorporated
3M Foundation
©2005 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
1. What is the sum of all the integers between –12.1 and 3.3?
1. _________________
2. Triangle AXY is similar to triangle ZBC. If AX = 6 cm, ZB = 18 cm
and ZC = 63 cm, what is the length of segment AY, in centimeters?
2. _________________
3. This is a type of magic square. The
eight sums from the numbers in each
of the 3 rows, in each of the 3 columns,
and in each of the 2 main diagonals are
equal. What is the value of n?
3. _________________
n+1
1
n−1
3
2n − 9
n
n−3
n+2
2
4. The value of 2x + 10 is half of the value of 5x + 30. What is the
value of x?
4. _________________
5. Cylinder B’s height is equal to the radius of cylinder A and
cylinder B’s radius is equal to the height h of cylinder A. If
the volume of cylinder A is twice the volume
of cylinder B, the volume of cylinder A can be
written as Nπh3 cubic units. What is the value of N?
5. _________________
B
A
6. Steve goes on a 40-km bike ride. He covers the first half of the
distance averaging a speed of 15 km/hr. In order to average
20 km/hr for the entire trip, how many kilometers per hour must
his average speed be during the second half of the trip?
6. _________________
7. In Mrs. Marsh’s algebra class, 30 of the 36 students took the
Chapter 4 test and the class mean was 72%. The next day the
other six students took the test and their mean score was 78%.
What is the new class mean? Express your answer as a percent.
7. _________________
3
4(3 )
8. What value of N satisfies 3 3 = 2 N ?
(4 )
8. _________________
9. What is the sum of the positive factors of 48?
9. _________________
10. A cylindrical can that is six inches high has a label that is 60π square
inches in area and exactly covers the outside of the can excluding
the top and bottom lids. What is the radius of the can, in inches?
10. _________________
11. A puppy and two cats together weigh 24 pounds. The puppy and
the larger cat together weigh exactly twice as much as the smaller
cat, and the puppy and the smaller cat together weigh exactly the
same as the larger cat. How many pounds does the puppy weigh?
11. _________________
12. At 3:20, what is the measure in degrees of the lesser angle formed
by the hour hand and the minute hand?
12. _________________
©2005 MATHCOUNTS Foundation: 2005 State Countdown Round
13. What common fraction is one-third of the way from
number line?
1
4
to
1
3
on a
13. _________________
14. What is the remainder when 22005 is divided by 7?
14. _________________
15. A quarter weighs the same as two pennies. The quarters in one
pound of quarters have a total value of $25. How many dollars
would a pound of pennies be worth?
15. _________________
16. If 3 × 11 × 13 × 21 = 2005 + b, what is the value of b?
16. _________________
17. On the horizontal line y = 5, there are two points exactly 5 units
from the point (7, 1). What is the sum of the x-coordinates of these
two points?
17. _________________
18. Express
0.6
as a common fraction.
1.3
18. _________________
19. In the diagram, there are more than three triangles.
If each triangle has the same probability of being
selected, what is the probability that a selected
triangle has all or part of its interior shaded?
Express your answer as a common fraction.
19. _________________
20. A number is increased by 50% and then the result is decreased by
50%. What is the percent of decrease from the original number to
the final number?
20. _________________
21. The algebraic expression (3x – 2y)2 – x(x + 4y) can be written in
the form Ax2 + Bxy + Cy2. What is the value of the sum
A + B + C?
21. _________________
22. Solve the equation 27 = 3(9)x-1 for x.
22. _________________
23. Let x represent the sum of the first 25 positive odd integers. Let y
represent the sum of the next 25 positive odd integers. What is the
value of y – x?
23. _________________
24. A bag contains 2 red chips and 3 blue chips. Kay will take two chips
out of the bag at the same time. What is the probability that the chips
will be different colors? Express your answer as a common fraction.
24. _________________
25. What is the simplified value of
10!+ 11!+ 12!
?
10!+ 11!
26. The figure drawn is not to scale. Which of
the five segments shown is the longest?
25. _________________
26. _________________
©2005 MATHCOUNTS Foundation: 2005 State Countdown Round
27. What digit must be placed in the blank to make the four-digit
integer 20_7 a multiple of 11?
27. _________________
28. If b3 = 0.25, what is the value of b-3?
28. _________________
29. Twenty-four hamsters weigh the same as 18 guinea pigs. Assuming
all hamsters weigh the same amount and all guinea pigs weigh the
same amount, how many hamsters weigh the same as 24 guinea pigs?
29. _________________
30. A circle has a radius of three inches. The distance from the center
of the circle to chord CD is two inches. How many inches long is
chord CD? Express your answer in simplest radical form.
30. _________________
31. How many total days were there in the years 2001 through 2004?
31. _________________
32. When x = 2 and y = −2, what is the value of ?
32. _________________
33. Triangles BAD and BDC are right triangles
with AB = 12 units, BD = 15 units, and
BC = 17 units. What is the area, in square
units, of quadrilateral ABCD?
33. _________________
34. Six horses eat 12 bales of hay in 12 hours. At the same rate, how
many hours will 36 bales of hay last 12 horses?
34. _________________
35. A team averages 7 losses for every 13 games that it wins. If ties are
not permitted, what is the probability that the team will win its next
game? Express your answer as a percent.
35. _________________
36. What is the sum of the digits of the decimal expansion of
22005 × 52007 × 3 ?
36. _________________
37. In a right triangle with integral length sides, the hypotenuse has
length 39 units. How many units is the length of the shorter leg?
37. _________________
38. Ed takes five 100-point tests in his algebra class. He scores 87, 85
and 87 points on the first three tests. If the scores of his last two
tests differ by three points and he has a 90% average on his five
tests, what was his highest test score?
38. _________________
39. If x + y = 16 and x – y = 2, what is the value of x2 – y2?
39. _________________
40. If three blots equal four bleets, and five bleets equal six blits, then
what is the ratio of one blit to one blot? Express your answer as a
common fraction.
40. _________________
©2005 MATHCOUNTS Foundation: 2005 State Countdown Round
41. Which of 1120,000, 530,000 or 270,000 is the greatest value?
41. _________________
42. Let p(x) = 2x – 1, where x is a prime number. What is the sum of the
three smallest distinct values of p(x)?
42. _________________
43. The least common multiple of x, 10 and 14 is 70. What is the
greatest possible value of x?
43. _________________
44. Express
7 5
+ −1 as a mixed number.
5 7
44. _________________
45. How many positive integers less than or equal to 10,000 are
divisible by 2, 5 or 10?
45. _________________
46. What is the x-coordinate of the x-intercept of the line containing the
points (7, 4) and (6, 3)?
46. _________________
47. Cutting equilateral triangle BFC out of square ABCD
and translating it to the left of the square creates
the shaded figure ABFCDE. The perimeter
of square ABCD was 48 inches. What is the
perimeter, in inches, of ABFCDE?
47. _________________
48. The teacher asks Bill to calculate a – b – c, but Bill mistakenly
calculates a – (b – c) and gets an answer of 11. If the correct answer
was 3, what is the value of a – b?
48. _________________
49. James has a collection of 20 miniature cars and trucks. Cars make
up 40% of the collection. If James adds only cars to his collection,
how many cars must he add to make the collection 75% cars?
49. _________________
50. How many distinct three-digit positive integers have only odd digits?
50. _________________
51. When three fair 6-sided dice are tossed, what is the probability of
tossing at least one 6? Express your answer as a common fraction.
51. _________________
52. The first three shapes in a sequence are made of toothpicks of the same
length and shown below. The first shape is made of 6 toothpicks. If
the pattern of adding toothpicks to form another congruent hexagonal
region continues, how many toothpicks
are needed to build the 100th shape?
52. _________________
53. If x is positive and x2 = 729, what is the value of x?
53. _________________
54. The area of rectangle ABCD is four times the
area of square AEFD, as shown. Points A, E
and B are collinear. What is the ratio of the
perimeter of ABCD to the perimeter of AEFD?
Express your answer as a common fraction.
54. _________________
©2005 MATHCOUNTS Foundation: 2005 State Countdown Round
55. Every June 1, an ecologist takes a census of the number of wrens in
a state park. She noticed that the number is decreasing by 40% each
year. If this trend continues, in what year will the census show that
the number of wrens is less than 10% of what it was on June 1, 2004?
55. _________________
56. Lark has forgotten her locker combination. It is a sequence of three
numbers, each in the range from 1 to 30, inclusive. She knows that the
first number is odd, the second number is even, and the third number is
a multiple of 3. How many combinations could possibly be Lark’s?
56. _________________
57. A cube has a surface area of 216 square centimeters. What is the
volume of the cube, in cubic centimeters?
57. _________________
58. Suppose 2a − 3b = –23. Given that a and b are consecutive
integers, and a < b, what is the value of a?
58. _________________
59. The 5 a.m. temperatures for seven consecutive days were −7°,
−4°, −4°, −5°, 1°, 3° and 2° Celsius. What is the mean 5 a.m.
temperature for the week in degrees Celsius?
59. _________________
60. Using only the digits 7, 8 and 9, how many positive seven-digit
integers can be made that are palindromes?
60. _________________
61. Triangle ABC has sides of length 5, 12 and 13 units, and triangle
DEF has sides of length 8, 15 and 17 units. What is the ratio of
the area of triangle ABC to the area of triangle DEF? Express your
answer as a common fraction.
61. _________________
62. A line has a slope of –7 and contains the point (3, 0). The equation
of this line can be written in the form y = mx + b. What is the
value of m + b?
62. _________________
63. What is the least integer whose square is 48 more than its double?
63. _________________
64. The four partners in a business decide to split the profits of their
company in the ratio 2:3:3:5. If the profit one year is $26,000 what
is the largest number of dollars received by any of the four partners?
64. _________________
65. What is the ratio of the area of triangle BDC
to the area of triangle ADC? Express your
answer as a common fraction.
65. _________________
66. The first term of a particular sequence is 5, and each successive
term is one less than twice the previous term. What is the value of
the sixth term?
66. _________________
67. Express 0.63 as a common fraction.
67. _________________
©2005 MATHCOUNTS Foundation: 2005 State Countdown Round
68. In this diagram, both polygons are regular. What is the value, in
degrees, of the sum of the measures of angles ABC and ABD?
68. _________________
A
69. Rays BA and BC form an acute angle.
Including angle ABC, how many acute angles
are formed by the rays in the diagram?
69. _________________
G
F
B
E
D
C
70. Both the numerator and the denominator of the fraction are to
be increased by two. What is the positive difference between the
new fraction and the original fraction? Express your answer as a
common fraction.
70. _________________
71. The line joining (3, 2) and (6, 0) divides the
square shown into two parts. What fraction of
the area of the square is above this line? Express
your answer as a common fraction.
71. _________________
72. If x is a positive number, by how much does x + 10 exceed 10 − x?
Express your answer in terms of x.
72. _________________
73. If g(x) = 2x2 − 3 and h(x) = 4x3 + 1, what is the value of g(h(−1))?
73. _________________
74. Points P and Q are midpoints of two sides of the
square. What fraction of the interior of the square is
shaded? Express your answer as a common fraction.
74. _________________
75. What is the slope of a line perpendicular to the line containing
the points (−1, 2) and (1, −2)? Express your answer as a common
fraction.
75. _________________
76. For how many positive integer values of x is the sum x2 + 4x + 4
less than 20?
76. _________________
77. Triangle ABC is a right triangle. If the measure
of angle PAB is x degrees and the measure of
angle ACB is expressed in the form Mx + N
with M = 1, what is the value of M + N?
77. ________________
78. The number 64 is both a perfect cube and a perfect square, since
43 = 64 and 82 = 64. What is the next larger number that is both a
perfect cube and a perfect square?
78. _________________
79. What is the least possible positive difference between the sum of
five different positive integers and the greatest of them?
79. _________________
80. If each face of a cube has a perimeter of 12 centimeters, what is
the volume of the cube, in cubic centimeters?
80. _________________
©2005 MATHCOUNTS Foundation: 2005 State Countdown Round