2005
Chapter Competition
Sprint Round
Problems 1–30
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of 30 problems.
You will have 40 minutes to complete the problems. You
are not allowed to use calculators, books or any other aids
during this round. If you are wearing a calculator wrist
watch, please give it to your proctor now. Calculations
may be done on scratch paper. All answers must be
complete, legible and simpliﬁed to lowest terms. Record
only ﬁnal answers in the blanks in the right-hand column
of the competition booklet. If you complete the problems
before time is called, use the remaining time to check your
answers.
Total Correct
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
Scorer’s Initials
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Texas Instruments Incorporated
3M Foundation
©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
1. The chart below gives the air distance in miles between selected
world cities. What is the distance from Honolulu to Cape
Town?
Bangkok
Bangkok
Cape Town
Honolulu
London
6300
6609
5944
Cape Town
6300
11,535
5989
2. The pie chart shown represents a
survey of Canadians who do not use
the Internet. What is the percent of
Canadian non-users for whom cost is
not the primary barrier?
Honolulu
6609
11,535
7240
1. _________________
miles
London
5944
5989
7240
Primary Barrier
to Internet Access
%
2. _________________
Canadian Non-Users
3. It is now 12:00:00 midnight, as read on a12-hour digital clock.
In 122 hours, 39 minutes and 44 seconds the time will be A:B:C.
What is the value of A + B + C?
3. _________________
4. Shooting hoops for 30 minutes burns 150 calories. How many
calories would Kendra burn shooting hoops if she shot hoops
30 minutes every day for one week?
calories
4. _________________
5. Olga purchases a rectangular mirror
(the shaded region) that ﬁts exactly
inside a frame. The outer perimeter of
the frame measures 60 cm by 80 cm.
The width of each side of the frame
is 10 cm. What is the area of the
mirror?
sq cm
5. _________________
6. The product of two positive whole numbers is 2005. If neither
number is 1, what is the sum of the two numbers?
6. _________________
©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round
7. Three friends each ordered a large cheese pizza. Shauntee ate 23
26
of her pizza, Carlos ate 89 of his pizza and Rocco ate 27
of his
pizza. If the remaining portions from the three pizzas are put
together, what fraction of a large pizza do they make? Express
your answer as a common fraction.
7. _________________
8. It takes 24 minutes for Jana to walk one
mile. At that rate, how far will she walk in 10
minutes? Express your answer as a decimal to
the nearest tenth.
miles
8. _________________
9. Roslyn has ten boxes. Five of the boxes contain pencils, four of
the boxes contain pens, and two of the boxes contain both pens
and pencils. How many boxes contain neither pens nor pencils?
9. _________________
boxes
10. How many combinations of pennies, nickels and/or dimes are
there with a total value of 25¢?
combinations
10. _________________
11. What is the value of the following expression:
1 − 3 + 5 − 7 + 9 − ….. − 43 + 45 − 47 + 49 ?
11. _________________
12. A rectangular tile measures 3 inches by 4 inches. What is the
fewest number of these tiles that are needed to completely cover
a rectangular region that is 2 feet by 5 feet?
tiles
12. _________________
13. When plotted in the standard rectangular coordinate system,
trapezoid ABCD has vertices A(1, −2), B(1, 1), C(5, 7) and
D(5, 1). What is the area of trapezoid ABCD?
square units
13. _________________
©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round
14. Trey receives a 5% commission on every sale he makes. On the
sale of a $60 coat (before any discounts), how many more cents
will he receive if his commission is based on the original price of
the coat rather than the price of the coat after a 20% discount?
cents
14. _________________
15. Five distinct points A, B, C, D and E lie on a line, but not
necessarily in that order. Use the information below to determine
the number of units in the length of segment DC.
• E is the midpoint of segment AB.
• D is the midpoint of segment AE.
• Both C and E are the same distance from B.
• The distance from D to B is 9 units.
units
15. _________________
16. In the 19th century, Britain used a money system which included
pence, farthings, shillings and pounds. The following conversions
were used:
4 farthings = 1 pence
12 pence = 1 shilling
20 shillings = 1 pound
How many total farthings were equivalent to 1 pound and 5 pence?
farthings
16. _________________
17. What is the sum of all the distinct positive two-digit factors of 144?
17. _________________
18. The points B(1, 1), I(2, 4) and G(5, 1) are plotted in the standard
rectangular coordinate system to form triangle BIG. Triangle BIG
is translated ﬁve units to the left and two units upward to triangle
B’I’G’, in such a way that B’ is the image of B, I’ is the image of I,
and G’ is the image of G. What is the midpoint of segment B’G’?
Express your answer as an ordered pair.
(
,
)
18. _________________
19. The positive difference of the cube of an integer and the square of
the same integer is 100. What is the integer?
19. _________________
©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round
20. A rectangular sheet of paper is folded twice and then cut, as
shown below. All fold lines are dashed, and the portion that is to
be cut away is shaded.
20. _________________
Which of the following drawings (A, B, C, D, E or F) shows
what the paper looks like when it is unfolded after the cuts?
21. Henry took ﬁve tests, and his average score was 57 points. He
scored at least 50 points on each test. There were 100 points
possible on each test. What is the highest score that Henry could
have earned on any of the ﬁve tests?
points
21. _________________
22. How many ordered pairs (x, y) satisfy BOTH conditions below?
Condition I: x = 1 or y = 0 or y = 2
Condition II: x = 0 or x = 2 or y = 1
ordered pairs
22. _________________
23. Zan has created this rule for generating sequences of whole
numbers.
If a number is 25 or less, double the number.
If a number is more than 25, subtract 12 from it.
For example, if Zan starts with 10, she gets the sequence 10, 20,
40, 28, 16, … . If the third number in Zan’s sequence is 36, what
is the sum of the four distinct numbers that could have been the
ﬁrst number in her sequence?
23. _________________
24. Reverse the two digits of my age, divide by three, add 20, and
the result is my age. How many years old am I?
years
24. _________________
©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round
25. The sequence of integers in
the row of squares and in each
of the two columns of squares
form three distinct arithmetic
sequences. What is the value
of N?
25. _________________
26. To be able to walk to the center C of a circular fountain, a repair
crew places a 16-foot plank from A to B and then a 10-foot plank
from D to C, where D is the
midpoint of AB . What is the
area of the circular base of the
fountain? Express your answer
in terms of π.
square feet
26. _________________
27. The function f is deﬁned by f (n) = f (n − 1) + f (n − 2). It is also
true that f (1) = 3 and f (3) = 10. What is the value of f (6)?
27. _________________
c 2
ad
a 3 b 8
= , = and = , what is the value of 2 ? Express
b 4 c 9
d 3
b
your answer as a common fraction.
28. If
28. _________________
29. A play has two male roles, two female roles and two roles that can
be either gender. Only a man can be assigned to a male role, and
only a woman can be assigned to a female role. If ﬁve men and six
women audition, in how many ways can the six roles be assigned?
ways
29. _________________
30. What is the arithmetic mean of all of the positive two-digit integers
with the property that the integer is equal to the sum of its ﬁrst digit
plus its second digit plus the product of its two digits?
30. _________________
©2004 MATHCOUNTS Foundation: 2005 Chapter Sprint Round
2005
Chapter Competition
Target Round
Problems 1 and 2
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of eight problems,
which will be presented in pairs. Work on one pair of
problems will be completed and answers will be collected
before the next pair is distributed. The time limit for each
pair of problems is six minutes. The ﬁrst pair of problems
is on the other side of this sheet. When told to do so, turn
the page over and begin working. Record your ﬁnal answer
in the designated space on the problem sheet. All answers
must be complete, legible and simpliﬁed to lowest terms.
This round assumes the use of calculators, and calculations
may also be done on scratch paper, but no other aids are
allowed.
Total Correct
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
Scorer’s Initials
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Texas Instruments Incorporated
3M Foundation
©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
1. Francisco starts with the number 5, doubles it, adds 1, doubles
the result, adds 1, doubles the result, adds 1, and continues this
pattern of two alternating calculations. Phong, meanwhile,
starts with 5, adds 1, doubles the result, adds 1, doubles the
result, adds 1, doubles the result, and continues this pattern
of two alternating calculations. They each do eight total
calculations. What is the positive difference of their ﬁnal
results?
1. __________________
2. All vertices of this cube will be colored such that no two
vertices on the same edge of the cube are the same color. What
is the minimum number of colors that will be needed to color
the vertices of this cube?
colors
2. __________________
©MATHCOUNTS Foundation: 2005 Chapter Target Round
2005
Chapter Competition
Target Round
Problems 3 and 4
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
Total Correct
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
Scorer’s Initials
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Texas Instruments Incorporated
3M Foundation
©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
3. A store purchases televisions from a factory for $87.89 each.
The store normally sells one of these televisions for 225% of
the factory cost, but a store coupon gives 25% off this selling
price. Ignoring tax, how much does a customer with this
coupon pay for the television? Express your answer in dollars
to the nearest hundredth.
3. __________________
$
4. The symbols
represent four
different integers from 1 to 9. Using the equations below, what
is the value of
?
4. __________________
©MATHCOUNTS Foundation: 2005 Chapter Target Round
2005
Chapter Competition
Target Round
Problems 5 and 6
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
Total Correct
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
Scorer’s Initials
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Texas Instruments Incorporated
3M Foundation
©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
5. What is the greatest whole number that MUST be a factor of
the sum of any four consecutive positive odd numbers?
5. __________________
6. In the ﬁgure below, the smaller circle has a radius of two feet
and the larger circle has a radius of four feet. What is the total
area of the four shaded regions? Express your answer as a
decimal to the nearest hundredth.
sq feet
6. __________________
©MATHCOUNTS Foundation: 2005 Chapter Target Round
2005
Chapter Competition
Target Round
Problems 7 and 8
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
Total Correct
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
Scorer’s Initials
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Texas Instruments Incorporated
3M Foundation
©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
7. Jamie has a jar of coins containing the same number of nickels,
dimes and quarters. The total value of the coins in the jar is
$13.20. How many nickels does Jamie have?
nickels
7. __________________
8. John, Mike and Chantel will divide a pile of pennies amongst
themselves using the following process:
pennies
8. __________________
The number of pennies in the pile is counted.
• If the number of pennies in the pile is even, Mike will get
half of the pile.
• If the number of pennies in the pile is odd, one penny will
be given to Chantel, and John will get half the pennies
remaining in the pile.
This process is then repeated until the pile is empty. How
many pennies will Mike have at the end if the original pile
contains 2005 pennies?
©MATHCOUNTS Foundation: 2005 Chapter Target Round
2005
Chapter Competition
Team Round
Problems 1–10
School
Team
Members
, Captain
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of ten problems
which the team has 20 minutes to complete. Team
members may work together to solve the problems. Team
members may talk during this section of the competition.
This round assumes the use of calculators, and calculations
may also be done on scratch paper, but no other aids are
allowed. The team captain must record the answers on
his/her problem sheet, and all answers must be complete
and legible. Only the team captain’s problem sheet will be
scored.
Total Correct
Founding Sponsors
CNA Foundation
National Society of Professional Engineers
National Council of Teachers of Mathematics
Scorer’s Initials
National Sponsors
ADC Foundation
General Motors Foundation
Lockheed Martin
National Aeronautics and Space Administration
Texas Instruments Incorporated
3M Foundation
©2004 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
1. A competition problem requires one hour to fully develop
(write, proofread, edit, and typeset). This problem is then
given to 30,000 students, each working an average of 24
seconds to solve the problem. What is the ratio of a problem’s
development time to the total time spent by the students
to solve the problem? Express your answer as a common
fraction.
1. _________________
2. Select any three-digit multiple of 3. Calculate the sum of
the cubes of the digits of that number. This is now your new
number. Now calculate the sum of the cubes of the digits
of this new number. Continue this procedure of adding the
cubes of the digits of the resulting number until you arrive at a
number that is equal to the sum of the cubes of its digits. What
is this number?
2. _________________
3. Emma and Ed walk into a room containing 30 assembled
Tworks and enough pieces to assemble 100 more Tworks. A
Twork takes eight minutes to assemble
and 10 minutes to disassemble. If Emma
starts assembling Tworks as Ed begins
disassembling the ones that were already
made, and they both continue to work until
there are exactly 35 assembled Tworks in
the room (and no partially assembled or
disassembled Tworks), how many minutes
will Emma have worked?
minutes
3. _________________
4. How many triangles are in the ﬁgure below?
triangles
4. _________________
5. How many integers are solutions to the equation
integers
5. _________________
(x − 2 )(
25 − x 2
) =1 ?
©MATHCOUNTS Foundation: 2005 Chapter Team Round
6. The minute hand of a clock measures 10 cm from its tip to
the center of the clock face, and the hour hand from its tip to
the center of the clock face is 5 cm. What is the sum of the
distances, in meters, traveled by the tips of both hands in one
24-hour period? Express your answer to the nearest thousandth
of a meter.
6. _________________
meters
7. These are two shoelace patterns for two identical shoes with
fourteen holes each. Assume that the holes form a rectangular
grid and that each hole is 1 cm from its nearest horizontal
and vertical neighbor-holes. Calculate the ratio of the total
length of the shoelace shown in Pattern #1 to the total length
of the shoelace shown in Pattern #2. Express your answer as a
decimal to the nearest hundredth.
7. _________________
Pattern #1
Pattern #2
8. A four-digit perfect square number is created by placing two
positive two-digit perfect square numbers next to each other.
What is the four-digit square number?
8. _________________
9. If Ella rolls a standard six-sided die until she rolls the same
number on consecutive rolls, what is the probability that her
10th roll is her last roll? Express your answer as a decimal to
the nearest thousandth.
9. _________________
10. A standard deck of playing cards with 26 red cards and 26
black cards is split into two piles, each having at least one card.
In pile A there are six times as many black cards as red cards.
In pile B, the number of red cards is a
multiple of the number of black cards.
How many red cards are in pile B?
red cards
10. _________________
©MATHCOUNTS Foundation: 2005 Chapter Team Round