Workout 1
1. ________ Five friends are standing in a line. Cliff is standing directly behind Danny, and there
are two people between Cliff and Mark. Mark is standing somewhere behind Eric, but
somewhere in front of Tom. Which of the five friends is fourth in line?
points During a basketball game, four players scored all the points. Joe scored half the
2. ________
points, Frank scored one-third of the points, Ken made one three-point shot and Mike
scored his only point with a free throw. How many points were scored in the game?
$
3. ________
Lotta Dough used a $100 bill to pay for two books that cost $11.98 each, two videos
that cost $14.49 each and miscellaneous school supplies that cost $24.17. If the tax
rate is 7.25%, how much does Lotta receive in change from her $100?
people The circle graph shows the results of a 2002
4. ________
survey of 4050 people. How many people chose
Billy Crystal as the best Oscars host? Express
your answer to the nearest hundred.
5. ________ On Marika’s 8th birthday, in 2004, her father said, “My age is now four times your
age.” In what year will Marika’s father be able to say, “My age is now three times your
age,” on Marika’s birthday?
miles If Andy bikes at a steady rate of 15 miles per hour, how many miles will he bike in four
6. ________
minutes?
cans The number of cans in the layers of a display in a supermarket form an arithmetic
7. ________
sequence. The bottom layer has 28 cans; the next layer has 25 cans
and so on until there is one can at the top of the display. How
many cans are in the entire display?
% In 1993, there were 844,001 students who attended public school in Wisconsin. In
8. ________
2003, there were 880,301 students who attended public school in Wisconsin. What is
the percent of change in enrollment from 1993 to 2003? Express your answer to the
nearest tenth.
9. ________ A 2 by 3 by 4 rectangular prism is painted and then cut into
24 unit cubes. If a unit cube will be selected at random,
what is the probability that it will have fewer than two
painted faces? Express your answer as a common fraction.
$
10. _______
Eighteen acres of land sold for $27,766.80. At the same rate, what is the cost of six
acres of land? Express your answer to the nearest whole dollar.
MATHCOUNTS 2004-2005
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Workout 2
rects How many rectangles are in this figure? Each angle is a right angle.
1. ________
hits Cal Ripkin Jr. played in 2632 consecutive baseball games. During this time, he was at
2. ________
bat 10,221 times, and his batting average was .277, meaning that he got a hit 27.7% of
the times he was at bat. How many hits did Cal Ripkin Jr. get during this time period?
Express your answer to the nearest ten.
% A store puts everything on sale for 20% off. If the sales tax is 8%, what percent of
3. ________
the original marked price is the final cost including tax? Express your answer to the
nearest tenth.
$
4. ________
The house numbers in a housing development are the multiples of four starting at 4 and
ending at 120. If metal digits cost 50 cents each, what is the total cost of all of the
metal digits that are needed to number the houses in this development?
$
5. ________
The amount of money spent on each of the four days of Kia’s vacation is shown in the
bar graph below. What was the average amount of money spent per day of the
vacation?
dimes Susan has exactly $1.50 in change. She has 30 total coins selected from pennies,
6. ________
nickels, dimes and/or quarters. Half of the coins are nickels. There are two-thirds as
many pennies as there are nickels. How many dimes does Susan have?
revs A bike’s front wheel has a 1-foot radius. How many revolutions does the wheel make in
7. ________
a one-mile trip? There are 5280 feet in one mile. Use the p key on your calculator or
the approximation 3.142, and express your answer to the nearest whole number.
calories After 12 minutes on a stair-stepper, Rick has burned 210 calories. At the same rate,
8. ________
what is the total number of calories Rick will burn in a 20-minute workout?
% Today there were 4622 fans at the soccer game. Last week, there were only
9. ________
3095 fans at the soccer game. What is the percent increase in the number of fans
from last week’s game to today’s game? Express your answer to the nearest whole
number.
rows At the beginning of a program, the 105 members of a
10. _______
marching band stand in a rectangular formation named
Formation A. All of the band members then move into
Formation B, which is a different rectangular
formation with six more rows, but with two fewer band
members per row. How many rows are in Formation A?
Problem #6 submitted by Mathlete Katie Prater, Blacksburg, Va.
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Workout 3
integers How many positive three-digit integers are divisible by both 11 and 5?
1. ________
% Twenty-five teachers were asked the length, in minutes, of their daily drive to school.
2. ________
Their responses are recorded in this stem-and-leaf plot. What percent of the teachers
take 25 or more minutes to get to school?
Driving Time to School (Minutes)
0
1
2
3
4
5
0
0
1
1
6
1
2
2
3
6
2
3
2
7
8
4
2
9
9 9
5 7
2 8
terms The integer 49 can be written as the sum of smaller perfect squares in a variety of
3. ________
ways. One such way includes six terms: 25 + 9 + 9 + 4 + 1 + 1. If each term has a value
between 0 and 49, what is the fewest number of perfect-square terms that can be
added together for a sum of 49?
sq mi In 1999 there was an average of 6.27 people per square mile in Australia. The
4. ________
population of Australia at that time was 18,613,087 people. What was the area of
Australia? Express your answer to the nearest ten thousand.
miles Two cars leave Arlington on I-20, one traveling due east and one traveling due west.
5. ________
One car is moving at a rate of 37 miles per hour, and the other is traveling at a rate of
41 miles per hour. How many miles from each other will they be in four hours?
6. ________ The mean radius of Mars is 2106 miles and the mean radius of Pluto is 707 miles. The
volume of a sphere is πU with radius r units. Assuming that the planets
are perfect spheres, what is the volume of Mars divided by the volume of
Pluto? Express your answer to the nearest whole number.
$
7. ________
A coin machine keeps 8.9 cents per $1 of coins inserted. This ratio is also maintained
when only a fraction of a dollar is inserted. If Ashley inserts $50.50 worth of change
into the machine, how much money will the machine give her back?
feet Keli and Mario are planning to plant rectangular gardens
8. ________
of the same length, side by side with fencing all around
and dividing the two plots. The total amount of
fencing is 100 feet. If the total area of the two
plots is 336 square feet and the dimensions are
integers, what is the length of the fence that
divides the two plots?
times At sea level, air pressure is 14.7 pounds per square inch. If the pressure under water
9. ________
increases by 4.3 lbs/in2 for every 10 feet you descend, how many times greater is the
pressure at 400 feet below sea level than at sea level? Express your answer as a
decimal to the nearest tenth.
rots A circular disk has a radius of two units. A point is marked on the edge of the disk.
10. _______
The disk rotates about its center, causing the point to travel a distance of 90 units.
How many rotations did the disk make? Express your answer as a decimal to the
nearest tenth.
Problem #7 submitted by teacher Mark McMahon, Raleigh, N.C.
MATHCOUNTS 2004-2005
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Workout 4
people The graph below shows the top five vote-recipients from a poll of 5000 people at
1. ________
Westfield Middle School. To the nearest 10, how many people either voted for
Michael Jordan or Tiger Woods to endorse a product?
Percent of Votes
Who Would You Choose to Endorse a Product?
23.9
23.9
13.7
13.7
4.1
4.1
2.9
2.9
1.8
1.8
0LFKDHO
7LJHU
/DQFH
6KDTXLOOH
%UHWW
-RUGDQ
:RRGV
$UPVWURQJ
2
1HDO
)DYUH
2. ________ The digits of a two-digit number “AB” are reversed to form a second two-digit number,
and the lesser of the two-digit numbers is subtracted from the greater. What prime
number must be a factor of the difference if A ¹ B?
ft/sec The winner of the local triathlon swam 0.5 miles, biked 15 miles and ran 3.5 miles.
3. ________
He finished the race in 1 hour, 2 minutes and 2 seconds. For the entire triathlon,
what was his average speed in feet per second? Express your answer to the nearest
whole number.
$
4. ________
Mady’s Messenger Service had total sales of $10,000 this week. Studies have shown
that if Mady’s does not advertise during the week, the week’s sales total will be down
5% from the previous week’s sales total. If Mady’s does not advertise for the next
four weeks, what are the expected sales for the fourth week?
5. ________ If f (x ) = 8x 3 – 6x 2 + 12x – 9, what is the value of f (
)?
% “Buy 3, Get 2 Free!” is equivalent to purchasing the five items at a discount of what
6. ________
percent?
sq meters A circular field circumscribes a plot of land that is
7. ________
200 meters square, as shown. How many square meters of the
circular region are not inside the square region? Express your
answer to the nearest whole number. (Use the p key on your
calculator or the approximation 3.1416.)
units
8. ________
What is the perimeter of triangle ABC, shown to the left? Express
your answer to the nearest whole number.
miles George’s first trip was 300 miles. If on his second trip, the time necessary to complete
9. ________
the trip is doubled and his average speed is two-thirds the average speed of his first
trip, how many miles is this second trip?
cu ft In the book Holes, Stanley must dig a cylindrical hole that measures five feet across
10. _______
and five feet deep. What is the volume of a cylinder with these dimensions? Express
your answer to the nearest whole number.
Problem #3 submitted by teacher Mark McMahon, Raleigh, N.C.
MATHCOUNTS 2004-2005
43
Workout 5
shots Marika shoots a basketball until she makes 20 shots or until
1. ________
she has made 60% of her shots, whichever happens first.
After she has made 10 of her first 20 shots, how many
more shots in a row does she have to make to be finished?
2. ________ Five girls are each given two standard dice. Each of the girls rolls her pair of dice and
writes down the product of the two numbers that she rolled. What is the probability
that all of the girls have written down a product greater than 10? Express your
answer as a decimal to the nearest ten thousandth.
3. ________ What number is the reciprocal of 100p? Express your answer in scientific notation
rounded to three significant digits.
teams The four-member math team at Pecanridge Middle School is chosen from the math club,
4. ________
which has three girls and five boys. How many different teams made up of two girls
and two boys could be chosen?
sq cm In the diagram, ABCD is a square, AD = 30 cm, and AE = 34 cm.
5. ________
What is the area of trapezoid ABCE?
6. ________ If a set of seven positive integers has a mean of 5, what is the greatest possible
integer in the set?
7. ________ What is the sum of all the three-digit positive integers?
8. ________ During the school year, the 10 members of the chess club play a total of 900 games of
chess during their inter-club practice matches. Each member plays against every other
member N times. What is the value of N?
sq cm Segment AB measures 4 cm and is a diameter of circle P. In triangle ABC, point C is on
9. ________
circle P and BC = 2 cm. What is the area of the shaded region? Express your answer
as a decimal to the nearest tenth.
10. _______ Suppose 5 £ x £ 8. What is the greatest possible positive difference of the
expressions 3x - 4 and 5 - 6x ?
MATHCOUNTS 2004-2005
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Workout 6
1. ________ The letters P, Q, R, S and T have replaced digits in the equation P4Q + 2R5 = S1T.
Given that each digit 1 through 9 is used exactly once in the equation, which digit must
Q represent?
2. ________ The positive four-digit integers that use each of the four digits 1, 2, 3 and 4 exactly
once are ordered from least to greatest. What is the 15th integer in the list?
grams A 6-ounce can of tuna has 13 grams of protein. This is 23% of the
3. ________
recommended daily allowance (RDA) based on a 2000-calorie diet.
What is the number of grams of protein needed to get 100% of the
RDA for this diet? Express your answer as a decimal to the nearest
tenth.
buses The Ace Bus Co. has buses that hold 45 students and buses that hold 35 students. If
4. ________
they use the fewest possible number of buses to transport 500 students, and each bus
is filled to capacity, how many 45-passenger buses are used?
5. ________ Without overlap, place these seven tiles into the 4 by 4 grid so that the four-digit
number formed in the n th row equals the four-digit number in the n th column. The tiles
may not be flipped or rotated. What is the sum of the four-digit numbers in Rows 1
and 2? (Hint: Cut out the pieces and place them in the grid!)
ROW 1
ROW 2
6. ________ If the ten letters in MATHCOUNTS are arranged at random, what is the probability
that the word MATCH will be spelled out anywhere in the arrangement? Express your
answer as a common fraction.
7. ________ The cube of a positive integer n is 9 less than (n + 9)2. What is the value of n ?
cu feet A building has a rectangular foundation that is 20 feet by
8. ________
15 feet. The roof extends between two opposite walls, is
12 feet high at the peak and slants down to 10 feet at the
walls. What is the volume of the building?
9. ________ A positive, even four-digit integer contains only two different digits, and the sum of its
digits is 9. If this integer is exactly twice another number also satisfying these
conditions, what is the integer?
10. _______ Steve has three quarters, three nickels and three pennies. If Steve selects three coins
at random and without replacement, what is the probability that the total value is
exactly 35 cents? Express your answer as a common fraction.
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Workout 7
years In 1970, the median age of a man marrying for the first time was 23.2 years. By 1998,
1. ________
the median age rose to 26.7 years. Assuming that the same linear trend continues, what
will be the median age for a man marrying for the first time in 2005? Express your
answer as a decimal to the nearest tenth.
courses James wants to take summer classes but wants to go to class only between 9 a.m. and
2. ________
noon each day. The table below lists the available courses and class times. What is
the maximum number of different courses James can take?
Course
Archery
Archery
Cartooning
Diving
Ceramics
Ceramics
Days
MWF
TTh
MWF
MTWThF
MWF
MWF
Times (am)
9 - 9:50
9 - 10:20
11 - 11:50
11 - 11:50
10 - 10:50
11 - 11:50
Course
Karate
Karate
Drama
Physics
Math Magic
Days
MWF
TTh
MWF
TTh
TTh
Times (am)
9 - 10:20
9 - 10:50
9 - 9:50
8:30 - 9:50
10:30 - 11:50
mph Lisa flies from San Francisco to New York (2500 mi) in 5.5 hours and then flies from
3. ________
New York to Rome (3500 mi) in 7 hours. What is the average speed for her entire trip?
$
4. ________
Munchies Inc. charges $1.57 for a package containing a dozen of Otto’s favorite
cookies. Crunchies Inc. charges 60 cents for a package containing five of the same
cookies. What is the least amount of money Otto could spend to purchase five dozen
cookies?
pounds Corn costs 99 cents per pound, and beans cost 45 cents per pound. If Shea buys
5. ________
24 total pounds of corn and beans, and it costs $18.09, how many pounds of corn did
Shea buy? Express your answer as a decimal to the nearest tenth.
6. ________ The product of two positive three-digit palindromes is 436,995. What is their sum?
cu inches Take two sheets of 8½” by 11” paper. Fold one sheet vertically into fourths to form the
7. ________
sides of a rectangular prism. Fold the other sheet horizontally into fourths to form the
sides of a different rectangular prism. How much more volume than the smaller prism
does the larger prism have? Express your answer as a decimal to the nearest tenth.
8. ________ Michael will flip a coin nine times. What is the probability that or more of the flips
will be heads? Express your answer as a decimal to the nearest thousandth.
9. ________ If 888x + 889y = 890 and 891x + 892y = 893, what is the value of x - y ?
ways In how many ways can Harold, Steve, John, Roslyn, Marian and Connie line up so that
10. _______
no two of the three boys are next to each other?
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Workout 8
sq meters The measures of the length and width of a
1. ________
rectangular garden are each an integer. Ophelia
fences the perimeter of the garden and also
builds two divider fences, as shown.
Numerically, the total length of fencing required in meters and the area of the garden
in square meters are equal. What is the greatest possible area of this garden?
2. ________ On the last day of each calendar month, Sue pays off $10 of her credit card balance.
However, if there is still a balance remaining after this payment, a $2 service fee is
added to the remaining balance. On April 30, Sue paid off $10 of her $100 balance.
If Sue never makes another charge to her card, at the beginning of what month will
her balance first be below $70?
tones A clock on the mantle strikes one tone at 1:00 (a.m. or p.m.), two tones at 2:00, three
3. ________
tones at 3:00 and so on. The first tone of the day is the first of 12 tones at midnight.
How many tones will the clock strike in the month of January?
4. ________ A right triangle has sides measuring 2x – 1, 3x – 13 and 3x – 4 units. What is the value
of x ?
cubes If this pattern of cubes is continued, how many cubes will be in the 20th formation?
5. ________
Formation
1
Formation
2
Formation
3
6. ________ The value of 5x – x 5 is 61,318, and x is a positive integer. What is the value of x ?
7. ________ The sum of N positive integers is 17. What is the largest possible product these N
integers could have?
inches
8. ________
Right triangle ABC is divided, as shown, such that angle BAC is
bisected by $' . What is the length of $' ? Express your answer
as a decimal to the nearest hundredth.
meters The area of a rectangular plot is 48,000 square meters. If Kim walks along the
9. ________
diagonal from one corner to the opposite corner instead of walking along the edges,
she walks 120 meters less. What is the length of the diagonal?
10. _______ A box contains six cards. Three of the cards are black on both sides, one card is
black on one side and red on the other, and two of the cards are red on both sides. If
you pick a card at random from the box and see that the side facing you is red, what is
the probability that the other side is red? Express your answer as a common fraction.
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Workout 9
cu cm A right prism has a regular hexagonal base and has a height of 10 cm. The side length
1. ________
of the base is 4 cm. What is the volume of the prism? Express your answer in simplest
radical form.
sq feet Brandon wishes to fence in a rectangular area of his lawn for his rabbit. If the
2. ________
measure, in feet, of each side of the enclosure is a positive integer and the perimeter
of the enclosure is 70 feet, what is the positive difference between the area of the
largest possible enclosure and the area of the smallest possible enclosure he could
build?
3. ________ The sum of three consecutive integers is a multiple of 89 and is a value between 600
and 900. What is the sum of the three integers?
4. ________ Fred and Jane are playing a coin flipping game. The coin will be flipped. Fred will get
a point if it is a Head; otherwise, Jane gets a point. This process will continue and the
first person to score 10 points wins the game. The score right now is 8-7 in favor of
Fred. What is the probability Fred will win? Express your answer as a common
fraction.
Fred X
1 X
2 X
3 X
4 X
5 X
6 X
7 X
8 9 10
Jane X
1 X
2 X
3 X
4 X
5 X
6 X
7 8 9 10
5. ________ What is the positive three-digit integer that is four times the product of its digits?
bit str A bit string is a sequence of digits consisting only of 0s and 1s. How many 12-digit bit
6. ________
strings contain no consecutive 1s?
7. ________ Assume a , b and c are positive integers such that c 3 = a 2 – b 2. If c is the least odd
integer for which this is true, what is the least possible value of b ?
feet Planet Pemdas is a perfect sphere that does not
8. ________
rotate. If a rocket flies in a path following the
circumference of Pemdas, and maintains a constant
altitude of 25,000 feet above the surface for one
orbit, how many more feet than the planet’s circumference
is the path of the rocket? Express your answer to the nearest
thousand feet.
sq feet
9. ________
A regular octagon has sides of length 4 feet. If the midpoints of
alternate sides of the octagon are joined to form a quadrilateral, as
shown, what is the total area of the four shaded regions? Express
your answer as a decimal to the nearest tenth.
10. _______ A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three
jellybeans from the bowl at random and without replacement, what is the probability
that exactly two will be red? Express your answer as a common fraction.
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