Warm-Up 1
dolls Cara has 12 more dolls than Mara, Mara has twice the number of dolls as Sara, and
1. ________
Sara has 15 dolls. How many dolls do the three girls have together?
miles On their summer vacation, the Charen family flew from New York City to Raleigh and
2. ________
then to Atlanta on their way to New Orleans. Returning from New Orleans, they flew
to St. Louis and then to Detroit on their way to New York City. According to the
mileage chart below, what was the total length of the family’s round-trip?
Distance Between Cities (Miles)
Atlanta
Detroit
New Orleans
New York City
Raleigh
St. Louis
Atlanta
Detroit
732
466
866
400
547
732
1077
606
707
542
New
Orleans
466
1077
1300
869
674
New York
City
866
606
1300
496
938
Raleigh
St. Louis
400
707
869
496
828
547
542
674
938
828
-
throws Eleven girls are standing around a circle. A ball is thrown clockwise around the circle.
3. ________
The first girl, Ami, starts with the ball, skips the next three girls and throws to the
fifth girl, who then skips the next three girls and throws the ball to the ninth girl. If
the throwing pattern continues, including Ami’s initial throw, how many total throws are
necessary for the ball to return to Ami?
inches Marty is 64 inches tall, and Phillip is 68 inches tall. What is the arithmetic mean of the
4. ________
two heights?
triangles How many triangles are in the figure to the right?
5. ________
inches When you sketch a figure of an adult male, it is recommended that the head be of
6. ________
the total height. If the full drawn height of a 6-foot man is 10 inches, how many
inches should the drawing of his head be from top to bottom? Express your answer as
a decimal to the nearest hundredth.
mults How many multiples of 7 are between 30 and 790?
7. ________
cans Five aluminum cans can be recycled to make a new can. How many
8. ________
new cans can eventually be made from 125 aluminum cans?
(Remember that the first new cans that are made can then be
recycled into even newer cans!)
years I am 13 years old, and my coach is 31 years old, which is my age with the digits
9. ________
reversed. What is the fewest number of years in which the digits of our ages will be
reversed again?
times How many times does the digit 6 appear in the list of all integers from 1 to 100?
10. _______
Problem #9 submitted by Michael Viscardi, member of the 2003 National Champion California team.
MATHCOUNTS 2004-2005
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Warm-Up 2
units The perimeter of a rectangle is 12 units. The width of the rectangle is 9 units less than
1. ________
twice its length. What is the length of the rectangle?
mL An IV-bag with 1000 mL of fluid is delivering fluid to a patient at the
2. ________
rate of 3 mL per minute. After the first three hours, how many milliliters
of fluid will remain in the IV-bag?
integers For how many positive integers x is 100 £ x 2 £ 200 ?
3. ________
4. ________ What is the probability of Jonah picking a vowel if he randomly chooses a letter from
the word “CAT”? Express your answer as a common fraction.
tbsp If a certain recipe requires five tablespoons of flour for every two ounces of butter,
5. ________
how many tablespoons of flour are needed if two pounds of butter are used? There
are 16 ounces of butter in one pound.
meals At the mall’s food court, Crystal wants to buy a meal consisting of one entrée, one
6. ________
drink and one dessert. The table below lists Crystal’s favorite foods in the food court.
How many distinct possible meals can she buy from these options?
Entrées
Drinks
Desserts
Pizza
Chicken Teriyaki
Corn Dog
Fish & Chips
Lemonade
Root Beer
Frozen Yogurt
Chocolate Chip Cookie
sq units What is the area of the triangle with vertices (1, 4), (3, 1) and (11, 1)?
7. ________
bars A regular chocolate bar weighs 7 ounces. A jumbo chocolate bar weighs 5 pounds.
8. ________
How many regular chocolate bars would you have to eat in order to consume the same
total weight as a jumbo chocolate bar? Express your answer as a decimal to the
nearest tenth.
(
,
)
9. ________
What ordered pair of positive integers (m, n ) satisfies 7m + 12n = 43?
$
10. _______
A two-inch by six-inch board costs 24 cents per linear foot.
Jennifer needs to purchase 18 boards that are each
10 feet long. What is the total cost of the lumber?
MATHCOUNTS 2004-2005
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Warm-Up 3
% An automobile insurance company has compiled data from a survey of 1000 16-year-old
1. ________
drivers during the year 2003. According to the results below, what percent of them
have had at least two accidents? Express your answer to the nearest tenth.
Total #
of Accidents
# of 16-Year-Old
Drivers
0
1
2
3
4
124
234
346
176
120
cookies Connie spent the weekend making cookies. She made 60 sugar cookies, 80 chocolate
2. ________
chip cookies and 100 peanut butter cookies. She plans to make packages
of cookies that each contain an identical assortment of whole cookies.
How many cookies are in a package, assuming that she makes as
many packages as possible and uses all of the cookies she made?
sq units What is the greatest surface area of a rectangular solid that can be built from
3. ________
16 identical unit cubes?
4. ________ On a number line, the coordinates of P and Q are 8 and 48, respectively. The midpoint
of 34 is B, the midpoint of %4 is C, and the midpoint of 3& is D. What is the
coordinate of D?
5. ________ When n is an odd integer, there is a value of m for which n 2 + 3 = 4 ´ m. If n = 11,
what is the value of m ?
6. ________ What integer is closest to the value of 10p +
?
7. ________ What is the mean of the set {73, 78, 81, 90, 85, 97}?
calories Christine jogged for half an hour. Amy walked for 50 minutes. Using the information in
8. ________
this chart about exercise and calories burned, how many more calories than Amy did
Christine burn?
Activity
Calories Burned per 5 Minutes
Walking
28
Jogging
57
years Tyler’s age is half Mary’s age. In four years, Tyler’s age will be two-thirds Mary’s
9. ________
age. How old is Tyler now?
cm Quadrilateral ABCD is congruent to
10. _______
quadrilateral WXYZ. We are given that
AB = 5 cm, BC = 7 cm, YZ = 6 cm and
ZW = 4 cm. What is the perimeter of
quadrilateral ABCD?
MATHCOUNTS 2004-2005
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Warm-Up 4
students Today Darron’s teacher pairs each student with a partner to
1. ________
create exactly 12 pairs of students. Next week each student
will be paired with a different partner. Darron’s partner for
next week can only be chosen from how many students?
2. ________ Zan has created this rule for generating sequences of whole numbers:
If a number is 15 or less, triple the number.
If a number is more than 15, subtract 13 from it.
Therefore, if Zan starts with 10, she gets the sequence 10, 30, 17, 4, 12, …. If the
first number in Zan’s sequence is 34, what is the 8 th number in the sequence?
inches Isosceles trapezoid ABDC and rectangle CDEF are shown,
3. ________
with DE = 5 inches and CD = 12 inches. The area of
trapezoid ABDC is 50 square inches. What is BE?
factors How many positive factors does 48 have?
4. ________
5. ________ Two quantities a and b are said to vary inversely if the value of the product ab
remains constant. The number of questions, q, on a test varies inversely with the
number of points, p, each question is worth. If q = 20 when p = 5, what is the value of
q when p = 2?
digits How many digits does the smallest repeating block in the decimal expansion of 6. ________
contain?
7. ________ A pair of six-sided dice is rolled, and the sum is recorded. What is the probability
that this sum is a multiple of three? Express your answer as a common fraction.
% The chart below was created from the results of a radio station survey. What percent
8. ________
of the males surveyed listen to the station?
Listen to Station
Yes
No
39%
13%
29%
19%
68%
32%
Male
Female
Total
9. ________ What is the value of + fraction.
4 9 + 4 9
+
4 9 + 4 9
Total
52%
48%
100%
? Express your answer as a common
hours A bug crawls at a rate of 14p units per hour around a circle with a radius of 3.5 units.
10. _______
How many hours does it take the bug to complete 26 revolutions of the circle?
MATHCOUNTS 2004-2005
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Warm-Up 5
cm A set of magnetic strips and balls are connected in an alternating fashion, as shown.
1. ________
The length of the first four items in the chain is 14 cm, and the length of the first
seven items is 25 cm. What is the length of the first 25 items in the chain?
$
2. ________
At Martha’s Mothball Antiques, a group of shirts was not selling. Martha decided to
mark the price of each shirt down by 35%. The discounted price for each shirt is now
$26. What was the original price of a shirt?
3. ________ The year 2004 is a leap year. At the end of what date will one-third of the year have
elapsed?
4. ________
Michael spins the spinner twice. All three of the larger sectors
are equal in area and have central angles of 90°. The two
smaller sectors have equal area. What is the probability that he
will “WIN” on both spins? Express your answer as a common
fraction.
ft/sec Julia jogs at a rate of six miles per hour. What is her rate in feet per second?
5. ________
Express your answer as a decimal to the nearest tenth.
pencils Zach has three bags and a bunch of pencils to be placed into the bags. He is told to
6. ________
place the greatest number of pencils possible into each of the three bags while also
keeping the number of pencils in each bag the same. What is the greatest number of
pencils he could have left over?
% Kevin turns 3 years old on the day that Marta turns 7 years old. What percent of
7. ________
Kevin’s age is Marta’s age on the day when Marta turns 24 years old? Express your
answer to the nearest whole number.
numbers How many two-digit prime numbers have a units digit of 7?
8. ________
s
units The isosceles triangle and the square shown here have the
9. ________
same area in square units. What is the height of the triangle,
h, in terms of the side length of the square, s ?
h
lines Point A and line m are in the same plane, but A is not on m. How many lines containing
10. _______
A are parallel to m ?
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Warm-Up 6
1. ________ Nathan will roll two six-sided dice. What is the probability that he will roll a number
less than three on the first die and a number greater than three on the second die?
Express your answer as a common fraction.
sq in The shaded rectangular region covers one-fourth of the area of the
2. ________
large 10-inch by 16-inch rectangle. What is the number of square
inches in the area of the shaded region?
coins Tim has three times as many coins as Mike. If Tim gives one coin to Mike, Mike will
3. ________
have a total number of coins equal to half the number of coins Tim started with. How
many total coins do Tim and Mike have together?
$
4. ________ A wheel has a circumference of 3 meters. The radius can be expressed as
meters,
%π
with relatively prime integers A and B. What is the value of A + B?
5. ________ The n th triangular number is the number 1 + 2 + 3 + 4 + … + n. Thus, the third triangular
number is 1 + 2 + 3 = 6. What is the sum of the fourth and fifth triangular numbers?
digits When the expression 810 ´ 522 is multiplied out (written in decimal notation), how many
6. ________
digits does the number have?
sq cm The surface area of a sphere with radius r is 4pr 2. Including the area of its circular
7. ________
base, what is the total surface area of a hemisphere with radius 6 cm? Express your
answer in terms of p.
°F Celsius temperatures can be converted to Fahrenheit by doubling the Celsius
8. ________
temperature, reducing the result by 10%, and then adding 32. What Fahrenheit
temperature corresponds to 20 degrees Celsius?
9. ________ What is the value of x + y if the sequence 2, 6, 10, …, x, y, 26 is an arithmetic
sequence?
y
sq units The lines 2x + y = 6 and x + y = 10 together with the
10. _______
x-axis and y-axis are drawn on a coordinate grid to
form quadrilateral ABCD. What is the area of
quadrilateral ABCD?
x
MATHCOUNTS 2004-2005
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Warm-Up 7
$
1. ________
Jane spent the following amounts on her school lunches last Monday through Thursday:
$3.29 - Mon., $1.79 - Tues., $2.65 - Wed., and $3.29 - Thurs. After she bought her
lunch on Friday, she calculated that her average lunch cost for that week was $2.80
per day. How much did her lunch cost on Friday?
regions Four straight lines intersect a circular region. The lines and circle are coplanar, and
2. ________
two of the lines are parallel. What is the maximum number of non-overlapping regions
inside the circle?
3. ________ What is the digit in the tens place when 72005 is expressed in decimal notation?
4. ________ The product of the digits of a positive three-digit integer n is zero, the sum of n ’s
digits is 8, and n is an odd number. What is the arithmetic mean of the possible values
of n ?
cm Triangle ABC is isosceles with vertex angle B and legs %& and
5. ________
$% , as shown. What is the perimeter of triangle ABC?
% In a school of 250 students, everyone takes one English class and one history class
6. ________
each year. Today, 15 total students were absent from their English class and ten total
students were absent from their history class. Five of the students were absent from
both classes. If a student is chosen at random from this school, what is the probability
that s/he was not absent from either class? Express your answer as a percent.
quads Ten distinct points are identified on the circumference of a
7. ________
circle. How many different convex quadrilaterals can be
formed if each vertex must be one of these 10 points?
combos
8. ________
Thad has an unlimited supply of 3-cent and 4-cent
stamps. If he has to put exactly 37 cents of postage on
a letter, how many different combinations of 3-cent
and/or 4-cent stamps could Thad use?
9. ________ On a graph, a lattice point is an ordered pair (x, y ) with integers x and y. Exactly
15 lattice points lie strictly in the interior of the triangular region with vertices (0, 0),
(N, 0) and (N, N), where N > 0. What is the value of N?
10. _______ According to the linear function represented in this table, what is the value of x when
y = 8?
x
y
-4
23
1
20
6
17
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Warm-Up 8
1. ________ Rodney uses the following clues to try to guess a secret number:
It is a two-digit integer.
The tens digit is odd.
The units digit is even.
The number is greater than 65.
If Rodney guesses a number that has each of these properties,
what is the probability that Rodney will guess the correct number? Express your
answer as a common fraction.
points A basketball player averaged 17 points in his first eight games and 24 points in his next
2. ________
six games. What is his overall average for the 14 games?
3. ________ The binary operation # is defined as a #b = 2a – 3b. What is the value of
(4 # 5) # 6?
4. ________ If x + 5 < 8 and x is a prime number, what is the value of x ?
°
5. ________ In a convex heptagon, the degree measures of the interior angles are x, x, x – 2, x – 2,
x + 2, x + 2 and x + 4. What is the degree measure of the largest interior angle?
items Steve has one quarter, two nickels and three pennies. Assuming no items are free, for
6. ________
how many different-priced items does Steve have exact change?
inches A greeting card is six inches wide and eight inches tall. Point A
7. ________
is three inches from the fold, as shown. As the card is opened to
an angle of 45 degrees, through how many more inches than point
A does point B travel? Express your answer as a common
fraction in terms of p.
8. ________ What is the least positive multiple of 72 that has exactly 16 positive factors?
9. ________ John, Kevin, Larry, Mary and Naomi all volunteered to do some math tutoring. If their
teacher randomly chooses two of the five students, what is the probability of selecting
the two girls? Express your answer as a common fraction.
$
10. _______
Rosalee plans to open a savings account and a checking account. She has decided to
deposit a total of $45 per week, such that $20 goes into the checking account each
week and the remaining money goes into the savings account. When she has deposited
a total of $200 into her savings account, how much will she have deposited into her
checking account?
MATHCOUNTS 2004-2005
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Warm-Up 9
1. ________ What is the value of y in the arithmetic sequence y + 6, 12, y ?
2. ________ Bobby’s grade in math class is based on five test scores. He has scores of 73, 83 and
90 on his first three tests. What score will Bobby have to average on the last two tests
to get an overall average of exactly 80?
marbles Anna divides her collection of marbles into two equal piles.
3. ________
Her little sister then takes three marbles from one of the
piles, leaving 28 marbles in that pile. How many marbles were
in Anna’s original collection of marbles?
( , )
4. ________
A line containing the points (9, 1) and (5, 5) intersects the x -axis at what point?
combos Three darts thrown at this dart board land in three different regions. If the order
5. ________
that the three darts are thrown does not matter, how many
combinations of three different regions from the 10 possible
regions would result in a sum of exactly 50 points?
calendars A reference book lists a set of annual calendars. For any given year, there is a
6. ________
calendar in the set that corresponds to it. How many annual calendars must be
included in the set in order to have a corresponding calendar for every possible year?
minutes Kris starts his run at 7 a.m. He runs at a rate of 4 miles per hour. His course takes him
7. ________
three miles out and then back along the same path. John runs at a rate of 6 miles per
hour, starting at 7:10 a.m., and he runs the same course. How many minutes elapse
between the time John finishes and the time Kris finishes?
degrees In the regular octagon ABCDEFGH, what is the number of
8. ________
degrees in the measure of angle DAF?
9. ________ If x + y = and x – y =
common fraction.
, what is the value of x 2 – y 2 ? Express your answer as a
4
94
94
94
94
9
?
10. _______ When simplified, what is the value of MATHCOUNTS 2004-2005
45
Warm-Up 10
combos Nine Ping-Pong balls are numbered 1 through 9. How many different combinations of
1. ________
three balls have a sum of 16?
Gops If 2 Lops are equal to 4 Gops, then 3 Lops are equal to how many Gops?
2. ________
3. ________ One of the five faces of the triangular prism shown here will
be used as the base of a new pyramid. The numbers of
exterior faces, vertices and edges of the resulting shape (the
fusion of the prism and pyramid) are added. What is the
maximum value of this sum?
4. ________ What is the coefficient of x 3 when x 4 – 3x 3 + 5x 2 – 6x + 1 is multiplied by
2x 3 – 3x 2 + 4x + 7 and the like terms are combined?
5. ________ What is the smallest positive five-digit integer, with all different digits, that is
divisible by each of its non-zero digits?
6. ________ Marion’s first nine quiz scores are 77, 85, 79, 92, 86, 92, 76, 97 and 81. If the
scores of her next two quizzes change the mode of the 11 scores to a different single
value and decrease the mean, what is the difference between the greatest possible
mean of the last two quizzes and the least possible mean of the last two quizzes?
7. ________ Define A&B as $% =
1$ + %6 . What is the value of (3&5)&8?
triangles Darina has five sticks measuring 5 cm, 5 cm, 8 cm, 14 cm and 14 cm. Using exactly
8. ________
three sticks as the sides of a triangle, how many non-congruent triangles are possible
if the sticks are joined only at their endpoints?
9. ________ Ann starts counting the letters of the alphabet beginning with A. When she gets to Z,
she goes backwards from Y to A and then reverses again going from B to Z. If she
continues this process, what is the 2005th letter that she will count?
10. _______ When simplified, what is the value of
MATHCOUNTS 2004-2005
× + ÷ × − ?
47
Warm-Up 11
1. ________ There are six bottles of soda, three bottles of juice and one bottle of water in a
cooler. If a bottle is randomly selected from the cooler, what is the probability that it
is the bottle of water? Express your answer as a common fraction.
units The length of each side of an isosceles triangle is a prime number. The perimeter of
2. ________
the triangle is greater than 21 units and is not a prime number. What is the least
possible perimeter of the triangle?
3. ________ A three-digit integer is reversed to form another three-digit integer. The positive
difference between the two numbers is 396. What is the greatest possible value for
the original integer?
feet Luminary bags are equally spaced in a row along a
4. ________
straight, long road. There are 20 feet from the first bag
to the fifth bag, when measured as shown. How many feet
are there from the first bag to the 25th bag?
5. ________ In the following addition problem, each distinct letter represents a
different digit from 1 through 9. What is the greatest possible
value of the four-digit number represented by the word “EXAM”?
+
TAKE
HOME
EXAM
6. ________ An abundant number is a positive integer such that the sum of its proper divisors is
greater than the number itself. The number 12 is an abundant number since
1 + 2 + 3 + 4 + 6 > 12. What is the smallest abundant number that is not a multiple of 6?
sq cm The sum of the area of the six congruent circles is 150p square
7. ________
centimeters. What is the area of the rectangle?
8. ________ The product of three consecutive positive integers is equivalent to 33 times the sum of
the three integers. What is the sum of the integers?
inches Jack is tiling his patio with concrete pavers that have a pattern in which six congruent
9. ________
rectangles are arranged. One such paver is shown here. If the
area of the paver is 600 square inches, what is its perimeter?
arrs A parking lot has a row of eight parking spaces numbered sequentially 1 through 8.
10. _______
Four cars (red, white, blue and green) are parked such that no two cars are in
adjacent parking spots. One such arrangement is shown here. How many arrangements
are possible?
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Warm-Up 12
sq cm The lateral surface area of the frustum of a solid right
1. ________
cone is the product of one-half the slant height (L) and
the sum of the circumferences of the two circular
faces. What is the number of square centimeters in
the total surface area of the frustum shown here?
Express your answer in terms of p.
cans In a two-day period, a committee drank 14 cans of soda. How many cans of soda
2. ________
would the committee drink in a week if they continue drinking at the same rate?
3. ________ The function f (n ) = n 2 + n + 17 for 0 £ n £ 15 generates prime numbers. What is the
value of f (10) – f (9)?
degrees The time is 4:40 p.m. What is the degree measure of the smallest angle formed by the
4. ________
minute hand and the hour hand?
inches An 8” by 10” picture is placed on top of a larger rectangular
5. ________
mat so that there is a border of the same width along each
side of the picture. If the area of the border around the
picture is 88 square inches, what is the outside perimeter of
the entire mat?
triangles How many triangles are there whose three vertices are points on this square
6. ________
3 by 3 grid?
inches The students of Glenview Middle School are planning a carnival. One of the contests
7. ________
will be a modified dart throw. The radius of the entire round dartboard
(made of two concentric circles) is eight inches. What is the radius of
the shaded circle if the area of the non-shaded region is three times
the area of the shaded region?
8. ________ A magician designed an unfair coin so that the probability of getting a Head on a flip
is 60%. If he flips the coin three times, what is the probability that he flips more
Heads than Tails? Express your answer as a common fraction.
9. ________ There exists a two-digit integer, “AB,” such that AB = 2(A + B). What is the value of the
product A × B?
quarts A car radiator has a 16-quart capacity and is currently filled with a 40% antifreeze
10. _______
solution. How many quarts of this solution should be drained off and replaced with
100% antifreeze to obtain a 50% antifreeze solution? Express your answer as a
common fraction.
Problem #9 submitted by Mathlete Timothy Smith, Bolingbrook, Ill.
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Warm-Up 13
girls Out of 28 students in Mr. Sullivan’s homeroom, the ratio of boys to girls is 3:4. The
1. ________
ratio of students who have returned their field trip forms to those who have not is 3:1.
If exactly half of the boys returned their field trip forms, how many girls have not
returned their forms?
integers How many three-digit positive integers can be formed using only odd digits less
2. ________
than six?
3. ________ The sum of two numbers is 15, and their product is 16. What is the sum of the
reciprocals of the two numbers? Express your answer as a common fraction.
4. ________ A 10-inch by 12-inch picture is to be enlarged and painted on
a 15-foot by 10-foot wall. If the picture remains proportional
(not distorted) and maintains the same orientation (not
rotated), what is the greatest fraction of the wall that could
be covered by the painting? Express your answer as a
common fraction.
5. ________ What is the value of a if the lines 2y - 2a = 6x and y + 1 = (a + 6)x are parallel?
sq units A regular hexagon has a side length of 6 units. What is its area? Express your answer
6. ________
in simplest radical form.
sums How many distinct sums can be obtained by adding three different numbers from the
7. ________
set {-2, -1, 1, 2, 3, 4, 5}?
weights A two-pan balance scale comes with a collection of weights.
8. ________
Each weight weighs a whole number of grams. Weights can be
put in either or both pans during a weighing. To ensure any
whole number of grams up to 100 grams can be measured, what
is the minimum number of weights needed in the collection?
9. ________ Ann has the spinner shown below. The six sectors of the spinner are congruent. What
is the probability of spinning integers with a positive difference
of 1 on her first two spins? Express your answer as a common
fraction.
10. _______ Let a and b be two integers for which 7a + 12b = 1. What is the largest possible
positive value of a + b which is less than 2005?
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Warm-Up 14
cards Ten cards are numbered and lying face up in a row, as shown. David turns over every
1. ________
card that is a multiple of 2. Then he turns over every card that is a multiple of 3, even
if the card had been turned over previously and is currently face down. He continues
this process with the multiples of 4 through 9. How many cards are then face up?
2. ________ Milton spilled some ink on his homework paper. He can’t read the coefficient of x , but
he knows that the equation has two distinct negative, integer solutions. What is the
sum of all of the distinct possible integers that could be under the ink stain?
x2 +
x + 36 = 0
integers How many integers between 100 and 500 have at least two 3s as digits?
3. ________
4. ________ The average of two 2-digit positive integers is equal to the decimal number obtained by
writing one of the two-digit integers before the decimal point and the other two-digit
integer after the decimal point. What is the smaller of the two integers?
5. ________ What is the value of
+ + + + ?
6. ________ There are equal numbers of pennies, nickels, dimes and quarters in a bag. Four coins
are pulled out, one at a time, and each coin is replaced before the next is drawn.
What is the probability that the total value of the four coins will be less than
20 cents? Express your answer as a common fraction.
7. ________ The arithmetic mean of 10 consecutive even integers is 3. What is the least of these
10 even integers?
sq units In the figure shown, arc ADB and arc BEC are semicircles, each with a radius of
8. ________
one unit. Point D, point E and point F are the midpoints of
arc ADB, arc BEC and arc DFE, respectively. If arc DFE is
also a semicircle, what is the area of the shaded region?
9. ________ Sue owns 11 pairs of shoes: six identical black pairs, three identical
brown pairs and two identical gray pairs. If she picks two shoes at
random, what is the probability that they are the same color and
that one is a left shoe and the other is a right shoe? Express your
answer as a common fraction.
days Each student works at the same speed. If five students can complete a job in six days,
10. _______
how many days would it take three students to complete the same job?
Problem #8 submitted by Mathlete Linyi Gao, Moscow, Idaho.
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Warm-Up 15
ways Raquel colors in this figure so that each of the four unit squares is
1. ________
completely red or completely green. In how many different ways
can the picture be colored this way so that there is a horizontal
line of symmetry at $% ?
2. ________ A positive two-digit number is even and is a multiple of 11. The product of its digits is
a perfect cube. What is this two-digit number?
posts A fence encloses a triangular region with sides measuring
3. ________
30 feet, 20 feet and 20 feet. The fence posts are
placed 20 inches apart, as shown. How many fence
posts are needed to enclose the triangular region
with the fence?
$
4. ________
Bryce bought 32 stools that required assembly from Need-A-Seat. Some stools have
three legs, and the other stools have four legs. The box arrived with 108 stool legs.
If the four-legged stools cost $20 and the three-legged stools cost $15, how much did
all of Bryce’s stools cost?
hours One pump can empty a tank in eight hours. A second pump can empty the same tank in
5. ________
five hours. What is the positive difference between the time it would take the faster
pump to empty the tank working alone and the time it would take for the two pumps to
empty the tank working together? Express your answer as a common fraction.
6. ________ The value of y is positive for all x > a in the equation y = (2x –1)(4x 2 + 4x + 1). What
is the least possible value of a ? Express your answer as a common fraction.
7. ________ A collection of seven positive integers has median 3 and unique mode 4. If the
collection has two 2s added to it, the median and unique mode are then both 2. What
is the mean of the new collection? Express your answer as a common fraction.
8. ________ A box contains two coins with a Head on both sides, one standard coin and one coin
with a Tail on both sides. A coin will be randomly selected from these four coins and
will be flipped twice. What is the probability that each of the two flips will result in a
Head? Express your answer as a common fraction.
elements Set A has two more elements than set B, and set A has 96 more subsets than set B. How
9. ________
many elements are in set A?
(
,
) Triangle ABC has vertices A(1, 0), B(0,-2) and C(3, -3). If
10. _______
triangle ABC is rotated 90° counterclockwise about A, what
are the coordinates of the image of C?
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Warm-Up 16
1. ________ On a number line, point M is the midpoint of segment AB. The coordinates of A and M
are –2 and 7, respectively. What is the coordinate of point B?
numbers How many two-digit prime numbers less than 50 exist such that the difference of their
2. ________
digits is even?
feet A carpenter wants to make the largest possible circular
3. ________
tabletop from a 4-foot by 8-foot sheet of plywood. If
the tabletop is constructed from the two largest
congruent semi-circular pieces that can be cut from
the sheet, what is the diameter of the resulting table?
(Assume no wood is lost when cutting the plywood.)
4. ________ Suppose f (x ) = x 2 + 12. If m > 0 and f (3m) = 3(f (m)), what is the value of m ?
cm A piece of wire 180 cm long is cut into two pieces with integer lengths. Each of the
5. ________
two pieces is formed into its own square with integer side lengths. The total area of
the two squares is 1073 cm2. How many centimeters longer is a side of the larger
square than a side of the smaller square?
+
+
+ . Express your
6. ________ Calculate the sum of the geometric series +
answer as a common fraction.
sums Suppose x and y are two distinct two-digit positive integers such that y is the reverse
7. ________
of x. (For example, x = 12 and y = 21 is one such combination.) How many different
sums x + y are possible?
cm In rectangle ABCD, AB = 6 cm, BC = 8 cm, and DE = DF. The area
8. ________
of triangle DEF is one-fourth the area of rectangle ABCD. What
is the length of segment EF? Express your answer in simplest
radical form.
9. ________ Four packages are delivered to four houses, one to each house. If these packages are
randomly delivered, what is the probability that exactly two of them are delivered to
the correct houses? Express your answer as a common fraction.
rects Using the sides of the 64 unit squares of a standard chessboard, how
10. _______
many non-congruent rectangles can be formed? Two such rectangles
(1 × 4 and 4 × 3) are identified to the right.
Problem #3 submitted by coach Robert L. Cobb, Exmore, Va.
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Warm-Up 17
1. ________ The “roundness” of an integer greater than 1 is the sum of the exponents of the prime
factorization of the number. For example, 20 = 22 ´ 51, so 20 has a roundness of 3.
What is the roundness of 1,000,000?
values Sarah will have seven test scores. Each of the scores is an integer
2. ________
value. The first five scores are 82, 87, 92, 96 and 98. How many
distinct values are possible for the median of Sarah’s seven
scores once she takes the last two tests?
3. ________ What is the least positive integer n such that 3n – n 3 is an even four-digit number?
4. ________ What is the sum of all the positive two-digit integers divisible by both the sum and
product of their digits?
units A circle was drawn (to the right), and then these three unit squares
5. ________
were placed over the circle in order to keep the circle from view.
The three unit squares are joined such that point M is the midpoint
of side AB. What is the largest possible radius of the hidden
circle? Express your answer as a common fraction.
edges If the measure of an interior angle of a regular polygon is 162 degrees, and this
6. ________
polygon is the base of a prism, how many edges does the prism have?
7. ________ The following clues describe a list of five integers.
Two of the numbers are negative.
The median value is 8, the mean value is 18.6, and the range of the values is 60.
One of the numbers is a perfect square, and one of the numbers is one more
than a perfect square.
The difference between the least two integers is 6.
What is the greatest of the five integers in this list?
8. ________ What is the greater of the solutions to the equation x 2 + 15x – 54 = 0?
integers How many positive four-digit odd integers can be created using only the digits 0, 1, 2,
9. ________
3, 4 and 5 if repetition of digits is not allowed?
$
10. _______
An elf can make a doll in 20 minutes, and he can make a train in
15 minutes. The dolls and trains will be sold for charity; a doll will
make $7 profit, and a train will make $5 profit. If the elf must make
at least as many trains as dolls, what is the greatest possible profit
that can be made from the elf’s work during an eight-hour period?
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Warm-Up 18
students In a school of 100 students, 90 study English, 75 study Spanish and 42 study French.
1. ________
Every student must study at least one of the three languages. What is the least
possible number of students who could be studying all three languages?
units A circle is inscribed in an equilateral triangle. What is the side
2. ________
length of the triangle if the area of the circle is 75p square units?
combos How many combinations of two or more consecutive positive integers have a sum of 45?
3. ________
D
F
H
+
+
such that each variable is replaced by a
E
G
I
different non-zero digit and the value of the expression is 1. (The answer may not be
unique.)
4. ________ Rewrite the expression
sq units In rectangle CDEF, CD = 32 and DE = 24 units. In rectangle DFGH (which has not yet
5. ________
been drawn), FG = 12 units. What is the area of the intersection of the interiors of
the two rectangles?
6. ________ Tamyra is making four cookies and has exactly four chocolate chips. If she distributes
the chips randomly into the four cookies, what is the probability that there are no more
than two chips in any one cookie? Express your answer as a common fraction.
7. ________ If (x + y + z ) 4 is multiplied out and the like terms are combined, what is the
coefficient of the x 2yz term?
ways In how many ways can 81 be written as the sum of three positive perfect squares if the
8. ________
order of the three perfect squares does not matter?
9. ________ Twice a certain positive integer is equal to the sum of three consecutive squares.
Eleven times the same positive integer is equal to the sum of the next three consecutive
squares. What is the positive integer?
10. _______ Mandvil has one standard quarter and one special quarter with a Head
on both sides. He selects one of these two coins at random, and
without looking at it first, he flips the coin three times. If he flips a
Head three straight times, what is the probability that he selected the
special quarter? Express your answer as a common fraction.
Problem #9 submitted by coach Stanley Levinson, Lynchburg, Va.
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