Download 2005 - 2006 Warmups

Warm-Up 1
settings George is planning a dinner meeting for 150 people. Each table will be set for nine
1. ________
people. If there are enough tables to accommodate everyone, what is the fewest
possible number of unused table settings?
$
2. ________
Sami decided to park her car at Park&Go for her four-day trip. The cost is usually
$8 per day or partial day. However, Sami has a 20% discount coupon for her entire
bill. How much did Sami‛s parking cost?
3. ________ If x = 3, what is the value of 2x + 3?
Newborn Infant Weights (lbs, oz)
lbs
oz What is the median weight of the newborn
4. ________
infant weights represented in the stem-andleaf plot shown to the right?
(2 4 represents 2 lbs 4 oz.)
5
3 6 7 10
6
0 1 3 4 7 8 11 12 14
7
0 4 10 10 10 12 14
8
2 3 4 7 8 10
9
2 4 6
5. ________ Haley has enlarged a 3” by 5” picture so that both the length and width are tripled.
By what number must the original area of the picture be multiplied to get the
enlarged area?
bananas A shake from Jake‛s Shakes uses 2 tsp vanilla, 2 scoops of ice cream, 4 oz fresh
6. ________
banana and ½ cup milk. A banana (without the peel) weighs 6 oz on average. How
many whole bananas must Jake order so that he can make 500 shakes?
% The data in the graph below show the number of jobs in this week‛s classified ads
7. ________
that require various levels of math knowledge. Each job was placed in exactly one of
the four categories. What percent of the jobs needs either practical “shop” math
or arithmetic? Express your answer to the nearest whole number.
Number of Jobs
Mathematics Required for Different Jobs
25
25
20
15
22
16
20
10
5
0
Theoretical Applied
Math
Math
“Shop”
Math
Arithmetic
weeks Game birds such as ducks and pheasants that are raised on a farm need a brooder
8. ________
house temperature of 95°F to start. The temperature is then decreased five
degrees per week until the outside temperature is reached. How many weeks would
it take for the brooder house temperature to reach 60°F?
9. ________ What is the value of 5 (3 · 2) − 3 (2 · 5) ?
% Tara rolls three standard dice once. What is the probability that the sum of the
10. _______
numbers rolled will be three or more? Express your answer as a percent.
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Warm-Up 2
16 cm
cm A regular octagon has the same perimeter as the regular hexagon shown
1. ________
here with side length 16 cm. How long is each side of the octagon?
2. ________ What is the smallest prime factor of 1821?
marbles Three bags each contain the same number of marbles. (The weight of an empty bag
3. ________
would not affect the weight on the
scale.) If three bags of marbles
plus two more marbles balance
14 marbles, as shown, how many
marbles are in each bag?
factors A certain number n has factors of 15 and 10. What is the fewest total number of
4. ________
positive factors that n can have?
5. ________ Jacque and Emily work in a shoe store. Emily was trying to carry too many boxes and
dropped all of them, spilling four different pairs of shoes
into a pile on the floor. Jacque picked up one shoe from
the pile, and then Emily picked up a shoe from the pile.
What is the probability that the shoe Emily‛s holding does
not belong in the same box as the shoe Jacque‛s holding?
Express your answer as a common fraction.
faces How many total faces does a trapezoid-based prism have?
6. ________
% The odds of an event happening successfully are expressed as a :b , where a :b is the
7. ________
ratio of the probability of a successful event to the probability of an unsuccessful
event. The odds of rolling a 5 on an unfair die are 1:3. What is the probability of
rolling a 5 with this die? Express your answer as a percent.
8. ________ In this maze you may move only in a downward direction. Starting with 0.4 at
the top, you must continue to add each new number you land on to the sum of the
previous numbers you‛ve been on. Your
0.4
trail is complete when you land on and
add the last value, which is 0.4 at the
0.1
0.7
0.3
bottom. What is the least possible sum
for a complete trail through the maze?
Express your answer as a decimal to the 0.8 0.9 0.7 0.6 0.5 0.6 0.3
nearest tenth.
0.6
0.3
0.7
0.4
meters2 The area of a square garden plot is 64 square meters. When the length is doubled
9. ________
and the width is increased by three meters, how much area is gained?
feet During a severe thunderstorm, Tim saw a flash of lightning and then six seconds
10. _______
later heard a clap of thunder. The speed of sound is 1088 ft/sec. How far is Tim
from the lightning? Express your answer to the nearest thousand feet.
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Warm-Up 3
$
1. ________
Last month Sue made $200 by baby-sitting for 12 hours and tutoring for 20 hours.
This month she worked the same number of hours at each job as she did last month,
but increased her baby-sitting fee by $2 per hour and her tutoring fee by $1 per
hour. How much did she earn this month?
$
3. ________
This graph shows the raise schedule for Mr. Bright‛s
employees. If the linear pattern continues, what is the
amount of the raise for six years of service?
Amount of Raise
lines How many lines of symmetry does the symbol Ω have?
2. ________
$4000
$3500
$3000
$2500
$2000
$1500
$1000
$500
$0
1 2 3 4 5 6
Years of Service
4. ________ A British television program shows homes that are for sale. One home is listed as
costing 500,000 pounds, or $850,000. The ratio of pounds to dollars used in this
TV program can be written in the form “1 pound : x dollars.” What is the value of x ?
Express your answer as a decimal to the nearest tenth.
5. ________ What is the sum of the first 10 terms of the arithmetic sequence 1, −1, −3, … ?
mm Kat has a collection of porcelain dolls that have heights, in millimeters, of 24, 32, 24,
6. ________
27, 40, 36 and 34. What is the average height of the dolls?
7. ________ At an awards ceremony 280 personalized
certificates were given out in the
categories and quantities shown here.
Julie earned certificates in both fine
arts and mathematics. The principal
grabbed a certificate out of the stack
at random after all of the science and
social studies certificates were given
out. What is the probability that he
pulled out one of Julie‛s certificates?
Express your answer as a common
fraction.
Type of Award
Number of Awards
Mathematics
35
Science
38
Social Studies
29
Business
40
Physical Education
23
English
25
Fine Arts
30
Foreign Language
60
% A restaurant mixes two gallons of milk containing 1% fat and three gallons of milk
8. ________
containing 2% fat. What is the percent of fat in the mixture, expressed to the
nearest tenth?
turns In a particular dice game a player rolls two dice. The player can choose either the
9. ________
sum or the product of the numbers rolled for her score for that turn. To win the
game, a player must get a total score of exactly 102 points from the sum of her
scores. What is the least possible number of turns needed to win the game?
diags How many diagonals does a regular octagon have?
10. _______
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Warm-Up 4
units Using the grid below, follow the clues to find the location of the treasure.
1. ________
y
Start at (−1, −1). Follow the line y = x until x = 4.
Reflect this new point over the x-axis.
From this new point add seven to the y-coordinate
and subtract five from the x-coordinate.
This is the location of the treasure.
y=x
x
How far is the treasure point from the starting point?
$
2. ________
Jane bought six items from a basket of 25-cent and 50-cent items. Kyle bought six
items from a basket of 50-cent and 75-cent items. What is the difference between
the lowest possible cost to Jane and the highest possible cost to Kyle?
4c
m
cm The two triangles shown are similar. What is
4. ________
the perimeter of the larger triangle?
5 cm
8c
m
3. ________ Roman numerals are letters that represent numbers. Some of the most common
ones are I = 1, V = 5, X = 10, L = 50 and C = 100. The Roman system represents the
number 3 as III and 30 as XXX. If a symbol with a lower value is placed in front
of a symbol with a greater value, the lesser number is subtracted from the greater
number. For example, IV represents 5 – 1 = 4. What is the three-digit value of the
following subtraction problem: CCCXLVII – CLXVIII ?
10 c
m
7 cm
pairings Chandra has four bowls. Each one is a different color (red, blue, yellow, green). She
5. ________
also has exactly one glass the same color as each bowl. If she chooses a bowl and a
glass from the cupboard, how many pairings are possible? One such pairing is a blue
bowl and a yellow glass.
1 inch
6. ________ One color printer produces 300 dots per linear inch while another can produce
600 dots per linear inch. If the two printers both print the same 8-inch by 10-inch
photograph, what is the ratio of the total number of dots produced
1 inch
by the first printer to the total number of dots produced by the
second printer? This figure shows what a printer using four dots
per linear inch would produce. Express your answer as a common
fraction.
degrees If the measure of angle C is double the measure of angle B,
7. ________
what is the measure of angle A in triangle ABC?
C
A
21
B
zeros How many zeros are there in the product of 24 and 55?
8. ________
9. ________ What is the greatest possible value of ( ) x ( ) + ( ) if the parentheses are filled
with 2, 4 and 12, and each of the three integers is used exactly once?
10. _______ What is the least positive integer value of x for which the inequality 3x > 2x + 1 is
true?
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Warm-Up 5
1. ________ The number 169 can be written as the sum of three positive perfect squares. What
is the greatest of these perfect squares?
Greatest Depth Attained by Submersibles
Number of Feet
vehicles The world‛s tallest mountain,
2. ________
measured from its base, is Mauna
Loa. Only 13,680 feet of this
33,356-foot mountain are visible
above the water; the rest of the
mountain is under the Pacific Ocean.
Of the seven submersible vehicles in
the graph, how many could get to the
bottom of Mauna Loa on the ocean
floor?
35790
36000
30000
24000
19700 19700
18000
13000
12000
6000
9775
3000
3300
e -1 na vin ow on ste
er
s
a Al
ph SRV Cy
p T - Ja Trie
s
e
y
D
e
D rgo
th
Ba
A
3. ________ If x = −2, what is the value of 2x − 7?
girls The ratio of boys to girls in a class of 35 students is 3:4. How many more girls than
4. ________
boys are in the class?
(x, y)
5. ________ On a coordinate grid, the triangle is translated from
position A to position B. What is the value of x + y ?
B
(2, 3)
(9, 4)
A
(0, 0)
(4, 0)
meters The diagonals of a rhombus measure 10 meters and 24 meters and are perpendicular
6. ________
bisectors of each other. What is the perimeter of the rhombus?
7. ________ The Fibonacci sequence is 1, 1, 2, 3, 5, 8, … . Each number after the first two
numbers is the sum of the preceding two numbers. What is the first perfect square
greater than 1 to occur in this sequence?
tpks How many toothpicks would you need in order to
8. ________
build the sixth figure in the pattern? The first
three figures of the pattern are shown here.
The first figure is made from four toothpicks.
9. ________ When four standard six-sided dice are rolled, what is the most likely sum of the
four numbers?
10. _______ A rectangular frame measuring 10” by 15” has five square
holes for photos. Four of the holes are each 2” by 2”, and
one hole is 3” by 3”. What portion of the frame‛s area was
not cut away for the photo holes? Express your answer as
a common fraction.
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Warm-Up 6
1. ________ The set {2, 3, 4, 5, 9} contains five of the factors of which one of the following four
integers?
480
1260 1890 3888
$
2. ________
The table below shows the average amount paid to winners of the Kentucky Derby
races from 1976 through 2004. For example, there were four years that
15 horses started the race, and the first-place winners for those years averaged
$527,175. For the 29 races shown, what is the average winnings amount associated
with the median number of starters?
No. of
Starters
Frequency
First Prize
No. of
Starters
Frequency
First Prize
9
1
$165,200
16
3
$688,467
10
1
$228,650
17
3
$680,600
11
1
$186,900
18
2
$1,299,900
13
3
$452,450
19
6
$777,750
14
1
$628,800
20
3
$605,967
15
4
$527,175
21
1
$317,200
3. ________ What is the 11th term in the geometric sequence 3,
answer as a common fraction.
−3 3 −3
, , ,
2 4 8
… ? Express your
4. ________ Goran has a standard deck of 52 cards. He considers an ace to have a value of 1, a
jack has a value of 11, a queen has a value of 12, a king has a value of 13, and all other
cards are their face value. What is the probability that a randomly selected card
will have an even value? Express your answer as a common fraction.
5.
sq ft
________
Jen and Jerry need to carpet their L-shaped room, having dimensions
10’
as shown. The carpet they chose is a solid color and is sold in rolls
4’
that are 12 feet wide. How many square feet of wasted carpet will
they have if the minimum length from the roll is used and only one
10’
seam is permitted?
12’
6. ________ What is the 100th term of the sequence 2, 3, 5, 6, 7, 10, 11, … which consists of all of
the positive integers that are neither perfect squares nor perfect cubes?
7. ________ What is the integer value of x that minimizes the absolute value of (6x 2 – 2x )?
ways Cheryl, Don, Oleg, Shandra and Hans each have exactly one pet. Their five pets are
8. ________
a ferret, parrot, dog, cat and fish, but not necessarily in that order. In how many
different ways can the five friends be paired with the five animals?
9. ________ The mean of a collection of five positive integers is 7. If the collection has a unique
mode, what is the greatest possible member of the collection?
10. Figure
_______ Maria sees these first four figures of a sequence in her textbook. The first figure
is made from 6 toothpicks. Maria has 41 toothpicks. What is the largest figure of
this sequence Maria can create if the sequence were to continue forever?
...
Figure 1
MATHCOUNTS 2005-2006
Figure 2
Figure 3
Figure 4
31
Warm-Up 7
cu ft Annie estimated that her household recycles five cubic feet of materials per
1. ________
collection day. There are 26 collection days per year in her town of 16,000
households. If every household is similar to Annie‛s, how much recycled material is
collected in one year in Annie‛s town?
sec The distance from the earth to the sun is 93,000,000 miles, and light travels at
2. ________
186,000 miles per second. How many seconds does light from the sun take to reach
the earth?
$
3. ________
Jorge bought a TV at a 15% discount. Later he found that there should have been
a 20% discount, so the store gave him the $17 he was owed. What was the price of
the TV before any discount?
4. ________ The whole numbers are arranged in an array as shown. If the pattern continues
going down one column and up the next, what is the number at the bottom of the
19th column?
1
8
9
16
...
2
7
10
15
3
6
11
14
4
5
12
13
patterns A bracelet is made by stringing together
5. ________
four beads. Each bead is either red or
R
green. How many different color patterns
are possible for the bracelet, where patterns
are considered the same if turning one will
produce the other, as shown here?
...
R
R
G
G
G
R
G
4
6. ________ A rectangular 7-inch by 12-inch picture is framed with a
mat so that four inches of the mat are seen on the top and
on each of the sides. Five inches of the mat are seen on
the bottom. What fraction of the finished product‛s area
(picture and mat) is the picture? Express your answer as a
common fraction.
4
4
7
12
5
7. ________
1
3
6
10
The numbers 1, 3, 6, 10, … are called triangular
numbers, as shown geometrically here. What is the
20th triangular number?
8. ________ If a # b = a 2 + b and a @ b = b − a , what is the value of ((1 # 3) @ 2)?
9. ________ If y = 2x + 1, which of the following equations is true?
A) x = 2y + 1
B) x = (½)y + 1
C) x = (½)y − ½
D) x = (½)y + ½
yel cds A special deck of cards consists of green cards and yellow cards. The odds of
10. _______
randomly selecting a green card from the deck are 3:5. If there are 40 cards in the
deck, how many yellow cards are in the deck?
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Warm-Up 8
$
1. ________
A resort hotel charges a regular rate of $100 per night, but Wednesday nights
are $150, and Saturday nights and Sunday nights are each $200. What is the
price difference between the most expensive and cheapest possible stays of four
consecutive nights?
%
2. ________
There are three times the number of orange fish as blue fish in a tank
at the pet store, and there are no other fish. If Kaya randomly pulls
out one fish from the tank, what is the probability that it is orange?
Express your answer as a percent.
sq cm A large square region is divided into four congruent, non3. ________
overlapping square regions. If the perimeter of one of these small
square regions is 60 cm, what is the area of the large square?
4. ________
Mary will pick a positive integer less than 80 that is a multiple of 7. Susan will pick
a positive integer less than 80 that is a multiple of 9. What is the probability that
they both will pick the same number? Express your answer as a common fraction.
hours A certain amount of work can be done by seven identical machines in 30 hours. How
5. ________
long will it take 10 of these same machines to do the same amount of work?
cubes How many 1” cubes will pack into a rectangular box with dimensions 8” by 6” by 6.5”?
6. ________
:
a.m. Changee is scheduling
7. ________
use of the pool for the
2008 Summer Olympic
Games. He has decided to
allow 15 minutes between
events, and he will use the
historic average time for
each event rounded up to
the next whole minute for
the length of each event.
He needs to schedule
these 10 events, in order,
beginning at 8 a.m. At
what time will the men‛s
relay begin, according to
his schedule?
Event
Avge Time of Event
(min:sec)
Women‛s Indiv Freestyle
2:23
Women‛s Indiv Butterfly
3:20
Women‛s Indiv Backstroke
3:10
Women‛s Indiv Breaststroke
2:56
Women‛s Relay
9:27
Men‛s Indiv Freestyle
2:10
Men‛s Indiv Butterfly
3:01
Men‛s Indiv Backstroke
2:54
Men‛s Indiv Breaststroke
2:45
Men‛s Relay
8:15
°F Each degree in the Celsius temperature system is 9 of a degree in the Fahrenheit
8. ________
5
system, and 32°F = 0°C. What is the Fahrenheit temperature when the Celsius
temperature is 45°?
9. ________ What number should be added to both the numerator and the denominator of
get a fraction equivalent to 54 ?
1
5
to
cm A rectangle has a perimeter of 38 cm. Three-fourths of its width is equal to one10. _______
fifth of its length. What is the measure of the length?
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Warm-Up 9
inches The perimeter of a rectangle is 24 inches. The length is 8½ inches. What is the
1. ________
width? Express your answer as a mixed number.
$
2. ________
Mrs. Goodteacher is ordering supplies for her 28 students. The chart shows what
she needs and how these items can be purchased, only in the quantities listed,
through her available supplier. A gross is 12 dozen, and a pound is 16 ounces. What
is the minimum total cost for her supplies?
Item
Quantity Needed
Purchase Info
A
Each student needs 2 pieces
1 gross for $2.38
B
Every 2 students share 1 piece
1 dozen for $2.27
C
Each student needs 1 ounce
1 pound for $2
D
Every 4 students share 1 piece
$10 each
ways Four people are sitting around a table. A name tag is made for each person. In how
3. ________
many ways can these name tags be passed out so that every person gets the wrong
name tag?
flowers Every other time that Cheri Linn saw a friend today, she gave away half of
4. ________
her flowers. At the end of the day, she had five flowers left. How many
flowers did she start the day with if she saw eight friends?
pgs A recent autobiography used 2808 digits to number the pages of the book. How
5. ________
many numbered pages are in the book?
6. ________ A positive two-digit integer less than 50 with two different digits, both of which
are odd, is chosen at random. What is the probability that the number is prime?
Express your answer as a common fraction.
inches Points A, B, E and F are the corners of an 8½-inch by 11-inch sheet of paper.
7. ________
Corners A and B are folded as shown below so that they meet at point D. The paper
is then unfolded. In the same manner, corners E and F meet at point G. What is the
length of segment GD? Express your answer as a mixed number.
B
E
11”
A
B
C
F
A
B
D
E
G
A
D
F
squares What is the least number of additional squares that must be drawn to
8. ________
make this figure have a vertical line of symmetry and a horizontal line
of symmetry? The figure may not be rotated.
solns How many positive integer solutions does the inequality 2x + 9 > 4x + 1 have?
9. ________
10. _______ A sequence of 20 terms is formed by using the rule f (x ) = 4x − 1 for
x = 1, 2, 3, …, 20. What is the sum of the last eight terms?
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Warm-Up 10
coasters The box-and-whisker plot below shows the speed of the 25 fastest roller coasters in
1. ________
the world. What is the largest possible number of roller coasters that could have a
speed greater than 80 mph?
Speeds (mph) of the Fastest Roller Coasters in the World
74
75.5
80
94
120
sq cm Each curve in the logo is a semicircle with a
2. ________
radius of 8 cm. What is the area of the logo?
Express your answer in terms of π.
liters A 10% potassium chloride solution is needed for use in a lab. How much water must
3. ________
be added to dilute 300 liters of a 50% potassium chloride solution for the lab‛s use?
4. ________ The sum of the digits of a positive two-digit integer is 45 less than the integer.
What is the tens digit of the integer?
triangles How many non-congruent triangles can be drawn if each one must have one side of
5. ________
length four inches, one side of length five inches and one 90º angle?
diagonals A square has two diagonals, and a convex pentagon has five diagonals. How many
6. ________
diagonals does a convex decagon have?
$
7. ________
Connie has 53 the amount of money as Joe. If Joe
gives Connie $3, then Connie would have three times
the money Joe has. How many dollars did Connie have
originally?
20
points Vanessa takes a test worth 85 points. If the minimum percent required to earn an
8. ________
A is 90%, and no rounding can be done, what is the least number of whole points she
can earn on the test to receive an A?
students A college class has exactly enough students to form eight equal
9. ________
rows. On Monday a student is absent, and the professor is
able to seat the students into five equal rows. On Tuesday two
students are absent, and the professor can seat the students
into nine equal rows. What is the least possible number of students in the class?
10. _______ Bag A contains four yellow marbles, two green marbles and three blue marbles.
Bag B contains four blue marbles and four green marbles. If you choose one marble
from each bag, what is the probability that you end up with one blue marble and one
yellow marble? Express your answer as a common fraction.
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Warm-Up 11
1. ________ A flight from San Francisco to New York takes five hours, but the return flight
takes six hours. When it is 4 p.m. in New York, it is only 1 p.m. in San Francisco. The
flight leaves San Francisco at noon and arrives in New York at x o‛clock, and the
return flight the next day leaves New York at noon and arrives in San Francisco at
y o‛clock. What is the positive difference of x and y ?
pieces A 6” by 6” grid made from 1” pieces of wire glued together at their
2. ________
ends is shown here. How many pieces of wire would be required to
create a similar 10” by 10” grid?
3. ________ What is the least positive integer that has each of the first eight positive integers
as factors?
$
4. ________
Jackson and Lisa divide some money in a ratio of 3:2, respectively. If Lisa gives
Jackson $6, then Jackson will have twice as much money as Lisa will have. How much
money did Lisa have after the original division of money?
sq units The length of a rectangle is 2x + 5 units, the width is 4x + 15 units and the
5. ________
perimeter of the rectangle is 100 units. What is the area of the rectangle?
°
6. ________ The consecutive angles of a particular trapezoid form an arithmetic sequence. If
the largest angle measures 120°, what is the measure of the smallest angle?
voters Sandy originally beat Terry by 82 votes in a school election. In a
7. ________
re-vote, all of Terry‛s supporters voted for her again, but some
of Sandy‛s supporters switched to Terry. If Terry beat Sandy by
four votes in the re-vote, how many voters switched?
VOTES
8. ________ Which of the following graphs represents the solution to x 2 > −4x − 3?
A.
4
2
0
2
4
B.
4
2
0
2
4
4
2
0
2
4
D.
C.
4
2
0
2
4
4
2
0
2
4
E.
integers How many positive integers less than or equal to 250 are relatively prime with 250?
9. ________
10. _______ The positive integers 2 through 2006 have median 1004. If the odd integers are all
increased by 1 and the even integers are all decreased by 1, what is the new median?
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Warm-Up 12
1. ________ A particular positive three-digit integer is divisible by 5. The integer also is
divisible by 11. The sum of the three digits of the integer is 13. What is the
integer?
2. ________ A European train compartment has six seats. Four
of the seats are broken. Wilhelm needs to fill out
a form like the one here to indicate that there are
broken seats. If he randomly checks off four of
the seats in the diagram, what is the probability
that he marked the correct seats? Express your
answer as a common fraction.
Maintenance Report
Door
$
3. ________
At the fish market, a whole fish costs $4.20. If the body costs $1 more than the
tail, and the body costs $2 more than the head, how much does the head cost?
4. ________ What is the greatest integer value of n such that 635,040 is divisible by 2n ?
cu in A ream of paper consists of 500 sheets. If a standard box of 8½” by 11” paper holds
5. ________
exactly ten reams with no extra space, and a sheet of paper is .004 inches thick,
what is the box‛s volume?
6. ________ Four positive four-digit integers with a sum of 23,750 each contain the digit 1 in a
different position. If the 1s are all removed, the list of integers becomes 982, 829,
982, 298. What was the last four-digit number in the original list?
7. ________ Scott and Deric are playing the Shade Game. Two standard
dice are rolled, and the sum of the numbers shown is the
number of blank squares that must now be shaded in. It
is Deric‛s turn, and his grid is shown here. What is the
probability that Deric will win on this turn by rolling the
exact sum needed to shade the rest of the grid? Express
your answer as a common fraction.
8. ________ Given the function f (a , b , c , d ) = d ÷ a − a b c − b c , what is the value of
f (1, 1, 1, −1)?
9. ________ A positive integer is 24 less than half its square. What is the integer?
10. _______ A polygonal region is bounded by the
lines x = 0, x = 4, y = −1 and y = 4. The
region is then rotated about the y -axis to
form a solid. Which of these nets is the
approximate net of the solid formed? (The
nets are not drawn to scale.)
MATHCOUNTS 2005-2006
A.
B.
C.
D.
49
Warm-Up 13
1. ________ What is the maximum product of two integers whose sum is 18?
regions When two lines are drawn in a plane, they can create four non-overlapping regions.
2. ________
If five straight lines are drawn in the plane, what is the largest number of nonoverlapping regions they can create?
floor One elevator leaves the 1st floor of a building at exactly 9 a.m. It takes
3. ________
five seconds to travel between floors, stays 10 seconds at each floor and
stops at each floor on its way to the top. Another elevator leaves the 12th
floor (the top floor) at exactly 9 a.m., takes five seconds to travel between
floors, stays 15 seconds at each floor and stops at each floor on its way
to the bottom. What is the first floor on which both elevators will be
available at the same time?
sq cm
4. ________
26
cm
A right square pyramid with a base measuring 20 cm by 20 cm is
unfolded into this net shown. Each side of the net not connected
to the square measures 26 cm. What is the area of the net of the
square pyramid?
5. ________ A set of five different positive integers has mean 8 and median 8. What is the
greatest possible value one of the integers could have?
coins Sam and Sally each have a pocketful of U.S. coins. Sam has
6. ________
49 cents while Sally has 43 cents. Neither has a sufficient
number of any coin to exchange for a larger value coin.
For example, neither has five pennies to exchange for one
nickel. What is the least number of coins they could have if
they combine their coins?
7. ________ Russell rolls a pair of dice with his eyes closed. His sister tells him that the sum of
the dice is greater than or equal to nine. Knowing this information, Russell calculates
the probability that he rolled doubles. What is that probability expressed as a
common fraction?
8. ________ The numbers 3, a , b , c , d , 38, … form an arithmetic sequence. What is the value of
the sum b + c ?
sq units Each student in a class is asked to draw a rectangle with integral side lengths and a
9. ________
perimeter of 30 units. Each student then calculates the area of the rectangle s/he
drew. What is the greatest possible difference of the areas of two of the students‛
rectangles?
grams Four cashews and three walnuts weigh 10 grams; two walnuts and six pecans weigh
10. _______
13 grams; and one cashew, one walnut and two pecans weigh 6 grams. Assuming each
cashew has the same weight, each walnut has the same weight and each pecan has
the same weight, how much does one walnut weigh?
MATHCOUNTS 2005-2006
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Warm-Up 14
items Pens cost $2, and mechanical pencils cost $3. If Dominic spent exactly $100 on
1. ________
these items, what is the least number of items he could have bought?
sq cm The boundary of the figure to the right is comprised of eight
2. ________
semicircular arcs, each with a diameter of 7 cm. What is the area of
the enclosed region?
min Each cartoon on a certain DVD lasts eight minutes, and there is a ten-second break
3. ________
between every two consecutive cartoons. Including the breaks, how long will it take
to watch all ten cartoons? Express your answer as a mixed number.
runs The Monroe Monkeys scored in each of the odd-numbered innings of a 9-inning
4. ________
baseball game. The Freeport Ferrets scored in each of the even-numbered innings
of the game. The two teams never scored in the same inning. The Monkeys scored
consecutive even numbers of runs starting with two runs in the first inning. The
Ferrets scored consecutive odd numbers of runs starting with one run in the second
inning. By how many runs did the winning team win?
buses An Ausdal bus from Austin to Dallas leaves every hour on the hour. An Ausdal bus
5. ________
from Dallas to Austin leaves every hour on the half-hour. The trip between the two
cities takes three hours. How many Ausdal buses going in the opposite direction
does an Ausdal bus from Austin to Dallas pass?
$
6. ________
The chart here shows the
prices for different seattypes at a stadium for
professional soccer. If every
seat in the stadium is sold for
a game, what is the average
price per seat?
Section
Price
No. of Seats Avail.
West, center
$10
1000
West, end
$8
1500
East, center
$8
800
East, end
$6
1700
weeks Paitlyn estimates she will need $750 for the computer she wants. She baby-sits for
7. ________
the Knox family every Saturday for $5.50 per hour. On average, she baby-sits
4.5 hours each Saturday. What is the fewest number of weeks she must work in
order to save enough money for the computer?
8. ________ Three digits are randomly selected without replacement from two 2s, three 6s and
one 9. What is the probability that the sum of the three digits is a multiple of 2?
Express your answer as a common fraction.
9. ________ If the reciprocal of (x − 2) is (x + 2), what is the greatest possible value of x ?
Express your answer in simplest radical form.
sq cm In this figure of three non-overlapping regular polygons, the
10. _______
area of the equilateral triangle is 43 square centimeters.
What is the area of the hexagon? Express your answer as a
common fraction in simplest radical form.
MATHCOUNTS 2005-2006
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Warm-Up 15
min Alex can bike one lap of a racecourse in 30 minutes at 25 miles per hour How many
1. ________
minutes does it take Lance to bike the same lap at 30 miles per hour?
2. ________ A point (a , b ) is located in the third quadrant of the Cartesian plane. The
coordinates are changed according to the functions given: f (a ) = a 2 + 2 and
g (b ) = b − 4. In which quadrant is (f (a ), g (b )) located?
3. ________ The mean of a collection of five positive integers is 6. What is the greatest possible
median of the collection?
Start
4. ________ In the maze to the right, you may move only
vertically or horizontally. Each box you move
to must have a fraction that is a value less than
the fraction of the box you are currently in. If
you start at the indicated box, what fraction is
in the last box of your path on the bottom row?
34
8
20
5
21
10
4
2
32
8
25
5
5
2
9
3
24
5
22
4
12
5
7
3
23 18 9
6
2
4
11 18 20
3 5
5
24 20 33
8
6
9
8 34 15
2
9
8
5. ________ If 2x > 8000, what is the least possible integer value of x ?
13
1
2
3
4
5
6
7
8
9
10
11
12
14
15
16
17
18
19
20
•••
6. ________ The whole numbers are listed in order
and placed in this arrangement. Each
row has two more entries in it than the
row above it. What is the first number
in the row containing the integer 500?
sq units Quadrilateral ABCD is a rectangle, and its length is twice
7. ________
its width. The perimeter of the rectangle is 60 units.
Points P, Q, R and S are midpoints of each side. What is
the area of quadrilateral PQRS?
A
P
S
B
Q
D
R
C
cuts A deck of 52 playing cards is printed on a 4 by 13 rectangular grid. If stacking the
8. ________
pieces is allowed after each cut, but no folding is
allowed, what is the least number of cuts required
to separate all the cards? (Cuts can be any length,
but may go in only one direction per cut.)
units In triangle ABC, m ∠ A = 30° and m ∠ C = 105°. If AC = 8 units, what is the length of
9. ________
segment BC? Express your answer in simplest radical form.
10. _______ A bag contains six white marbles, five red marbles and four blue marbles. Norka
will reach into the bag and pull out three marbles at the same time. What is the
probability that all three marbles will be the same color? Express your answer as a
common fraction.
MATHCOUNTS 2005-2006
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Warm-Up 16
ways In how many ways can this figure be colored if the five regions
1. ________
must be colored with either red, white or blue and no two
bordering regions can be the same color?
2. ________ The lines y = ax + 1 and y = 1x + a , with a ≠ 1, intersect in exactly one point. What is
the x -coordinate of that point?
ways A band has four trumpets, three trombones, three saxophones and two drums. If
3. ________
each section has its own set of seats, in how many ways can the 12 band members be
seated?
trumpets
trombones
drums
saxophones
4. ________ Miss James has a license plate with six digits. If commas were
05 IDAHO 06
______
placed after the second and fourth digits, it would look like the
number is a list of three consecutive even numbers in increasing
order. The first of those numbers has three different prime factors. What is the
least six-digit number that can be on Miss James‛ license plate?
5. ________ When the least common multiple of 8 and 20 is multiplied by the greatest common
factor of 8 and 20, what is the result?
fractions Think about all of the positive common (reduced) fractions with a denominator of 11
6. ________
7
or less. How many of these common fractions have a value less than 11
?
7. ________ If 12x + 57 = 29, what is the value of 12 x + 92 ?
min Mr. Harter drives from Chicago to Milwaukee on Sunday. On Monday when he makes
8. ________
the return trip, he figures that if he increases his speed by one-fifth, he can make
the trip in 80 minutes. How many minutes did the first trip take?
sq cm The sphere has radius 10 cm, and the smaller circle‛s center
9. ________
is 6 cm from the center of the sphere, as shown in the figure.
What is the area of the smaller circle in terms of π ?
6
10
ways In how many ways can two slashes be inserted between the letters of
10. _______
MATHCOUNTS to create three groups of letters? Each “group” of letters must
contain at least one letter.
MATHCOUNTS 2005-2006
61
Warm-Up 17
Columns
1 2 3
4
Row 1
1
Row 2
2
3
Row 3
4
5
6
7
Row 4
8
9
10
11
5
6
...
12 13 14 15
...
1. ________ The positive integers are written
in this pattern where the number
of entries in each successive row is
twice the number of entries in the
previous row. In which column is the
number 2006?
2. ________ The sum of the first N positive integers is 2006. Wait, that‛s impossible... one of
those integers must not have been included! If in fact exactly one of the first N
positive integers was not included in the sum, which integer must that have been?
ways Each of the six identical faces of a cube is to be painted either gray or white. How
3. ________
many distinct cubes are possible? (Notice that the two cubes shown below are not
distinct. They are different orientations of the same cube.)
4. ________ The sum of the numerator and denominator of a fraction is 160. The fraction is
equivalent to 73 . What is the original fraction?
5. ________ Scott and Deric are playing the Shade Game. Three standard dice
are rolled, and the sum of the numbers rolled is the number of
blank squares that must then be shaded. It is Deric‛s turn, and
his grid is shown here. What is the probability that Deric will roll
the exact sum needed to shade the rest of the grid on this turn?
Express your answer as a common fraction.
6. ________ Ben owns five pairs of socks, and each pair is a different color. If he loses three
socks, what is the probability that he has only two complete pairs left? Express
your answer as a common fraction.
7. ________ Parallelogram ABCD has diagonals that intersect at point E. Suppose that
AB = 6x + 5, BC = 2x , AE = 4x − 3 and EC = 3x + 4. Is angle ABC an acute, right or
obtuse angle?
8. ________ The three integers e , f and g are distinct primes such that e = f 2 + g 2. What is the
greatest possible value of e less than 100?
9. ________ Nine points are to be evenly spaced on a standard number line. They are labeled A
through I, from left to right. The coordinates
A
I
of points A and I are 1 and 10, respectively.
What is the coordinate of point B? Express
10
1
your answer as a mixed number.
10. _______ Define the operation ◊ for any two real numbers a and b as a ◊b = 2a (a + b ). What
is the value of p such that 3◊(4◊p ) = 1122?
Problem #2 submitted by 2005 national competitor Noah Arbesfeld, Lexington, Mass.
MATHCOUNTS 2005-2006
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Warm-Up 18
1. ________ In the game “Dicey Numbers,” two dice are rolled, and the two numbers are used
to form a two-digit number. For example 6 and 1 form either the number 61 or the
number 16. The number is then changed as described below.
If your number is greater than 36, take 80% of the number and round to the
nearest whole number.
If your number is less than 15, take 120% of the number and round to the
nearest whole number.
If the original number was prime, add 5 to its current value.
If Sherry rolls a 1 and 4, what is the difference between the greatest final value
and least final value that could result?
tickets At a carnival game, Katya chooses a number from 1 through 6 and rolls two ordinary
2. ________
dice. If her number comes up on one die, she wins three tickets. If her number
comes up on both dice, she wins six tickets. If she plays all morning, how many
tickets would she expect to win, on average, per turn?
3. ________ The points (x, y) represented in this table lie on a straight line.
When the equation of this line is written in the form y = Ax + B,
what is the value of A + B?
x
y
2
t−2
t
7
v
v+6
4. ________ A non-zero fraction‛s denominator is increased by 1 and its numerator is decreased
by 50%. If the resulting fraction is 31 of the original fraction, what is the original
fraction‛s denominator?
°
5. ________ When the triangular piece of paper ABC is folded along the
dotted line AF, point B lands on side AC. We know AB is 1 unit
and AC is 2 units. If the measure of angle B is 90°, what is the
measure of angle AFC?
B
F
C
A
6. ________ Pat and Grady were both born on May 15. Pat was born in 1958, and Grady was born
in 1966. In what year did Pat‛s age become three times Grady‛s age?
%
7. ________
Carbohydrates provide four calories per gram, and fats provide nine calories per
gram. If 36% of the calories Curtis ate at lunch came from fats and 64% came
from carbohydrates, what percent of the grams eaten came from carbohydrates?
cards Jill gives half her cards to Bill; Bill then gives half of his cards to
8. ________
Jill. Jill and Bill repeat this two-step process twice more. If Jill
now has 37 cards and Bill has 19 cards, with how many cards did
Bill start?
ways The local sandwich shop has three cash registers. In how many possible ways can
9. ________
four customers line up at the registers?
cu cm A pyramid has a rectangular base 10 cm by 25 cm and four triangular faces; two
10. _______
opposing faces both with altitude 13 cm and the other two faces with altitudes of
20 cm and 15 cm. What is the volume of the pyramid?
MATHCOUNTS 2005-2006
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