CSU Math Day 2009 TEAM Competition
11:00
ANNOUNCEMENTS:
• This is a large school competition.
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. Simplify 01 + 10 .
ANSWER: 1
2. If Jenna’s shoebox contains 25 blue marbles, 25 green, and 50 red, what is the probability that a randomly selected marble will be red?
ANSWER: 50%
3. If the shortest leg of a 30-60-90 triangle has length 3, what is the sum of the three leg
lengths?
√
ANSWER: 9 + 3 3
4. Given a tetrahedron, what do you get if you start with the number of faces, subtract
the number of edges, and then add the number of vertices?
ANSWER: 2
5. An 18th Century Swiss mathematician is featured on Swiss currency, was honored on
stamps in three countries, and had an asteroid named after him. The previous problem
implicitly featured a formula of his, and he is responsible for much of the notation we
use today, including the letter e for the base of the natural logarithm. Who is he?
ANSWER: Euler
6. What is the volume of a sphere with radius 1?
ANSWER: 4π
3
11:00, page 2
7. What is the smallest positive integer whose square is more than 650?
ANSWER: 26
8. Free throws are 1 point baskets in basketball. Shelby makes 80% of the free throws
she attempts. Expressed as a decimal, how many points should she expect to score if
she attempts 8 free throws?
ANSWER: 6.4
9. The angles of a triangle sum to 180 degrees. The angles of a rectangle sum to 360
degrees. What is the sum of the degrees of the angles of a pentagon?
ANSWER: 540
10. By extending to projective space, how many times do parallel lines intersect?
ANSWER: Once
11. What is the next number in the following sequence, which was generated by a degree
2 polynomial? 1, 2, 4, 7, 11, 16.
ANSWER: 22
12. Each number in the Fibonacci sequence (after the first two) is formed by adding the
previous two numbers. Starting with 1 1, what are the first 8 Fibonacci numbers?
ANSWER: 1 1 2 3 5 8 13 21
TIEBREAKER: What is the area of a triangle with leg lengths 1, 1, and
ANSWER: 12
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
√
2?
11:00, page 3
Extra questions
1. The area of a square is 10 times the perimeter. What is the length of a side?
ANSWER: 40
2. A fair coin is tossed four times. What is the probability that heads appears at least
once?
15
ANSWER: 16
3. A list contains all whole numbers from 1 to 1001 exactly once. What is the mean of
the list?
ANSWER: 500.5
11:20
ANNOUNCEMENTS:
• This is a small school competition.
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. What is the greatest common divisor of 68 and 44?
ANSWER: 4
2. If you extend a square into a third dimension in the obvious way for an appropriate
distance, you get a cube. What do you get if you do the same thing with a circle?
ANSWER: A cylinder.
3. There is a 5% sales tax on the sale of all puppies. John’s labradoodle just had a litter
of 8 puppies. If each one sells for $200, how much total tax money will be collected?
ANSWER: $80
4. Projective space is the mathematical formalization of what artistic concept?
ANSWER: Perspective
5. What is the largest prime divisor of 262 − 52 ?
ANSWER: 31 (use difference of two squares factorization)
6. What is a 12-sided polygon called?
ANSWER: A dodecagon.
11:20, page 2
7. Matt can mow 1 lawn in 3 hours. Travis can mow 1 lawn in 6 hours. How long will it
take the two of them to mow 3 lawns if they work together?
ANSWER: 6 hours
8. How many prime numbers are there?
ANSWER: Infinitely many.
9. A book and a movie by the same name describe the life and work of a mathematician
who won a Nobel Prize. The book was named A Beautiful Mind. Name the mathematician.
ANSWER: John Nash.
10. If Tim gets $1 of allowance on day one and, from then on, gets an allowance each day
equal to double the previous allowance, how much money will he have after seven days?
ANSWER: $127
11. The length of a rectangle is twice the width. What is the area of the rectangle if the
perimeter is 36?
ANSWER: 72
12. Sofya rolls a pair of dice. What is the probability that the sum is 7?
6
ANSWER: 16 or 36
TIEBREAKER: Which is larger: 1002 or 2100 ?
ANSWER: 2100
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
11:20, page 3
Extra questions
1. What is the area of the triangle bounded by the lines x = 0, y = 0, and 2x + y = 6?
ANSWER: 9
2. A 6-foot tall man casts a 12-foot shadow. A flag pole next to him casts a 54-foot
shadow. How tall is the flagpole?
ANSWER: 27 ft
3. If I travel 1 mile west, 4 miles south, and 4 miles east. How far am I from my starting
point (assuming I am on flat land!)?
ANSWER: 5 miles
11:40
ANNOUNCEMENTS:
• This is a large school competition.
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. What is 14 in binary?
ANSWER: 1110 (14 = 8 + 4 + 2)
2. The sum of two integers is -9 and their product is -36. What is the larger of the two
numbers?
ANSWER: 3 (The other is -12.)
3. Who locked himself in his attic for eight years to solve Fermat’s Last Theorem?
ANSWER: Andrew Wiles
4. What is the next number in the following sequence, which was generated by a degree
2 polynomial? 2, 5, 10, 17, 26, ...
ANSWER: 37 (n2 + 1)
5. How many feet are there in 5 miles?
ANSWER: 26,400
6. What is the smallest angle of a triangle with sides 1, 2, and
ANSWER: 30 degrees or π6
√
3?
11:40, page 2
7. What is the surface area of a cylinder with radius 3 and height 5?
ANSWER: 48π
8. What is the smallest positive integer with 5 distinct prime factors?
ANSWER: 2310 (= 2 · 3 · 5 · 7 · 11)
9. If a cube has volume 343, what is its surface area?
ANSWER: 294
10. What is the last digit of 251 ?
ANSWER: 8
11. If A is 10% of B, B is 10% of C, and C is 10% of D, then what is D divided by A?
ANSWER: 1000
12. Whose triangle has a 1 on the top row, 1 1 on the next row, 1 2 1 on the next row,
and so on?
ANSWER: Pascal
TIEBREAKER: The face of a brick measures 7 inches tall by 14 inches long. How many
brick faces would you be able to see if standing in front of a wall that is 7 feet tall and 28
feet long?
ANSWER: 288
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
11:40, page 3
Extra questions
1. What is the vertex of the parabola with equation y = x2 + 4?
ANSWER: (0, 4)
2. Find the area of a triangle with side lengths 2, 3, and 5.
ANSWER: 0
3. If f (x) = x2 − 1, find f (f (f (−1))).
ANSWER: 0
12:00
ANNOUNCEMENTS:
• This is a small school competition.
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. What is the prime factorization of 36?
ANSWER: 2 · 2 · 3 · 3
2. A circle of radius 3 is inscribed in a square. What is the area of the space between the
square and the circe, in terms of π?
ANSWER: 36 − 9π
3. What is the next number in the following geometric sequence? 1, − 12 , 41 , − 18 .
1
ANSWER: 16
4. Two airplanes, piloted by Shawn and Elly, are flying in opposite directions pass each
other at 8:15 p.m., one flying at 300 mph and one flying at 500 mph. At what time
will they be 600 miles apart, assuming they change neither speed nor course?
ANSWER: 9:00 p.m.
5. There are 10 people in a room, all immune to swine flu. They each shake hands exactly
once with each other person. How many handshakes occur?
ANSWER: 45: They each shake hands with 9 people (90), but you have
double-counted each handshake.
6. Sarah knows that she needs six liters of paint to paint a cube. How much paint will
she need to paint a cube with twice the sidelength of the original cube?
ANSWER: 24
12:00, page 2
7. What research institute kicked off the 21st century by offering one million dollars to
the first person to solve any of seven key problems?
ANSWER: the Clay Math Institute (Clay is adequate)
8. If tan(θ) = −1√and cos(θ) is negative, what is sin(θ)?
ANSWER: 22
9. What is the sum of the first 5 prime numbers?
ANSWER: 28 (= 2 + 3 + 5 + 7 + 11)
√
10. A rectangle has length 5 3 and width 5. How long is a diagonal?
ANSWER: 10
11. Factor x3 + 23 x2 + 19 x.
ANSWER: x(x + 13 )2
12. What is the sum of the integers from 1 to 15, inclusive?
ANSWER: 120 (= 15 · 16/2)
TIEBREAKER: Shorty is 1 foot taller than Ricky and twice as tall as Bobby. How
tall is Ricky if Bobby is 3 feet, 9 inches tall?
ANSWER: 6.5 feet OR 6 feet, 6 inches OR 78 inches
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
12:00, page 3
Extra questions
1. Find the fourth vertex of a rectangle, three of whose vertices are (−4, 4), (−4, 0), and
(−8, 4).
ANSWER: (−8, 0)
2. If f (x) is linear with f (4) = 3 and f (−5) = −6, what is f (−4)?
ANSWER: −5
3. The average of 3 numbers is 28. What is the average of these 3 numbers together with
12?
ANSWER: 24
12:20
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. Patrick wants to pick one shirt, one hat, and one pair of pants to wear. If he has 4
shirts, 6 hats, and 2 pants, how many different outfits can he make?
ANSWER: 48 (= 4 · 6 · 2)
2. What do you get if you divide the area of a circle of radius R by the area of a square
with side length R?
ANSWER: π
3. What is the constant term of a polynomial with roots -2, 5, 3, 1, and 0?
ANSWER: 0
4. What kind of coordinates are used in projective space?
ANSWER: Homogeneous
5. What is the smallest positive integer greater than one that is a square, a cube, and a
fourth power?
ANSWER: 4096 (= 212 )
6. Ryan and Hilary are working on a beautiful sculpture. It is a perfectly proportioned
cylinder. Unfortunately, due to budget cuts, they cannot afford as much marble as
they had hoped. By what percentage is the volume of their cylinder decreased if they
halve both the radius and the height?
ANSWER: 87.5%
12:20, page 2
7. If Justin brushes his teeth for three minutes twice a day, every day for 70 years, to the
nearest multiple of 10, how many days will he spend brushing his teeth?
ANSWER: 100 or 102 (it’s actually 106.5)
8. Veteran’s Day is next week. It commemorates an armistice signed in November 1918.
How many calendar months ago was that?
ANSWER: 1092 (12 months per year for 91 years)
9. If i2 = −1, what is (1 − i)2 ?
ANSWER: −2i (= 1 − 2i + i2 )
10. What ancient Greek mathematician known as the “Father of Geometry” wrote the
textbook titled Elements?
ANSWER: Euclid
11. To the nearest positive integer, what is the fifth root of 33?
ANSWER: 2
12. Expressed as an integer, what is the product of the square root, cube root, and fourth
root of 4096?
ANSWER: 8192 (213 )
TIEBREAKER: Rachel has three options for her commute. Which of the following three
options should she expect will get her home fastest?
1. Driving 35 miles on a 70 mph highway where there is a 50% chance of a 40 minute
wait because of an accident;
2. Driving 34 miles on a 35 mph road which is guaranteed not to have any traffic; or
3. Driving 39 miles on a 50 mph road where there is a 20% chance of a 10 minute wait?
ANSWER: (c) (50 minutes vs. just under an hour vs. just under 50 minutes)
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
12:20, page 3
Extra questions
1. Find x so that the average of the four numbers 46, 31, 37, and x is 37.
ANSWER: 34
2. A shirt has been marked down 5% to $42.75. What was the original price?
ANSWER: $45
3. The largest of 6 consecutive integers is 105. What is the smallest of these integers?
ANSWER: 15
12:40
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. There are approximately 7 trillion bacteria on and in your body right now. (Gross,
huh?) How many zeros are in that number?
ANSWER: 12
2. What is the product of the first 6 primes?
ANSWER: 30030
3. Which 20th century mathematician is often credited with laying down the foundation
of abstract algebra, particularly during her years in Göttingen?
ANSWER: Emmy Noether
4. Give an equation of the form ax + by = c, with a, b, and c positive integers, for the
line through (4,2) and perpendicular to a line with slope 2.
ANSWER: x + 2y = 8, or any integer multiple
5. Express the following as an integer: ln(e12 ) − ln(e5 ).
ANSWER: 7
6. About 70 percent of our planet is covered with water. If a Cassie, and astronaut,
throws a stone at the Earth, what are the odds (NOT the probability) it will hit land?
3
or 30%!)
ANSWER: 3 to 7 or 37 (not 10
12:40, page 2
7. n factorial is the number n times n − 1 times “dot dot dot” times 2 times 1. For
example, 3! = 3 ∗ 2 ∗ 1 = 6. What is 7!?
ANSWER: 5040
8. What is 300!
299! ?
ANSWER: 300
9. Projective space plays a key role in what mathematical discipline?
ANSWER: Algebraic geometry.
10. What is the sum of the roots of x3 − 4x?
ANSWER: 0
11. The Large Hadron Collider is a massive circular particle accelerator near Geneva,
Switzerland. It has a circumference of 27 kilometers. To the nearest integer, how
many kilometers would Dr. Whitfield need to walk to get from a town in the center of
the circle to the collider at the edge of the circle?
ANSWER: 4
12. What is the sum of the integers from 7 to 20, inclusive?
ANSWER: 189 (= 20 ∗ 21/2 − 6 ∗ 7/2)
TIEBREAKER: Groups of cicadas emerge from the ground every several years and sunflowers have a certain number of swirling patterns in their seeds. Though these numbers
vary from insect group to insect group and from plant to plant, they all come from some
famous sequence. Name that sequence.
ANSWER: The Fibonacci sequence.
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
12:40, page 3
Extra questions
1. The price of a gold watch has been reduced by 20% and then 40%. The watch finally
sold for $384. What was the original price?
ANSWER: $800
2. What number is halfway between 81 and 67 ?
55
ANSWER: 112
3. Given 6 gallons of a 20% antifreeze/water mixture, how many gallons of pure antifreeze
must be added to yield a 70% mixture?
ANSWER: 10
1:00
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. How many diagonals are there in a 12-sided polygon?
ANSWER: 54
2. What is the positive square root of 5184?
ANSWER: 72
3. What famous mathematician kicked off the 20th century by putting forth 23 unsolved
problems at the 1900 International Congress of Mathematicians in Paris?
ANSWER: David Hilbert
4. Cayla took three exams and scored 80, 82, and 98. If these three exams and the final
are the only grades in the course and are equally weighted, what will Cayla need to
score on the final to achieve a course grade of 90?
ANSWER: 100
5. What is the technical name for the following number, given in scientific notation: one
times ten to the one hundred?
ANSWER: A google.
6. The graph of a degree 3 polynomial with leading coefficient 2 crosses the x-axis at 0,
2, and 4. What is the coefficient of the x2 term?
ANSWER: -12
1:00, page 2
7. Eric is an eccentric musician. He gathers 26 friends and has them say their ABCs in
a round. In the first second, the first person says A. In the second second, the first
person says B while the second person says A. Each second, another person begins
their alphabet. How many seconds elapse before the 26th friend is done?
ANSWER: 51
8. How many numbers between 1 and 100 are prime?
ANSWER: 25
9. Jaime went to the store to buy some flowers. On the way home, she gave half to her
mom, half of what remained to her sister, and half of what remained after that to her
niece. When she got home, she had only 3 flowers left. How many did she buy?
ANSWER: 24
10. (read slowly) Ellen warned me that the world will end in 2012. Here’s more evidence:
What is the result when you start with 2012, multiply by the first digit of 2012, add
the second digit of 2012, multiply by the third digit of 2012, and then divide by the
final digit of 2012?
ANSWER: 2012 (Oh dear – it’s true!)
11. (READ SLOWLY!) If a tub fills at a rate of 10 cubic inches per second and drains at
a rate of 5 cubic inches per second, how many seconds will it take to completely fill an
empty tub if the drain is left open and the tub is a cube with side length 10?
ANSWER: 200 seconds OR 3 minutes, 20 seconds
12. How many digits are in the product 251 · 549 ?
ANSWER: 50
TIEBREAKER: How many 7-digit phone numbers are there, assuming any string of 7
digits is a valid phone number?
ANSWER: 107 OR 10 million
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
1:00, page 3
Extra questions
1. An elevator moves at a constant speed and takes 24 seconds to go from the first floor
to the fifth floor. How long should it take to go from the first floor to the tenth floor?
ANSWER: 54 seconds
2. A bag of chicken feed will feed 25 chickens for 30 days. For how many days will it feed
15 chickens?
ANSWER: 50
3. How many distinct complex roots does the polynomial x6 + 1 have?
ANSWER: 6
1:20
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. Let x be a positive integer written in binary. What is the first digit of x?
ANSWER: 1
2. Factor the polynomial x3 − x2 − 6x into linear factors.
ANSWER: x · (x − 3) · (x + 2)
3. A certain type of cell divides every 5 minutes. Starting with 1 cell, how many cells will
there be at the end of one hour?
ANSWER: 4096
4. According to a 2008 Money Magazine survey and the latest U.S. News rankings, there
are only four top 100 schools for mathematics located in the top 25 places to live.
Three of them are Rutgers in Piscataway, NJ; UC Irvine in Irvine, CA; and University
of Oklahoma in Norman, Oklahoma. What is the fourth?
ANSWER: Colorado State, right here (shameless plug)
5. Suppose f (x) = 2x − 1. What is f (f (f (3)))?
ANSWER: 17
6. If each tree has 40,000 leaves and each leaf takes up 0.1 cubic centimeters of an 800
cubic centimeter bag, how many bags do you need in order to bag up all of the fallen
leaves from the 5 trees in your yard?
ANSWER: 25
1:20, page 2
7. How many digits are there in 1012 − 107 ?
ANSWER: 12
8. Projective space can be thought of as the space of lines through which special point?
ANSWER: The origin.
9. A number is called perfect if it is equal to the sum of its factors (other than itself).
For example, 4 is not perfect, because 1 + 2 = 3, not 4. What is the smallest positive
perfect number?
ANSWER: 6
10. For which values of a does ax2 + 6x + 3 have no real roots?
ANSWER: a > 3.
11. Nine people are seated in nine seats around a circular table. What is the largest number
of people that can be chosen so that no two of them sit beside one another?
ANSWER: 4
12. Nathan and Ashley are kind of strange. They never park their Ferrari unless the
odometer reading is a palindrome. When they got in their car this morning to go to
work, the odometer read 124421. What is the minimum distance they will need to
drive before parking again, assuming they go at least one mile?
ANSWER: 1100 miles
TIEBREAKER: Cheerleaders build pyramids with one person at the top, two in the next
layer, and one more in each subsequent layer. How many cheerleaders would be needed to
build a pyramid with 10 layers?
ANSWER: 55
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
1:20, page 3
Extra questions
1. Mack, Billy, and Buddy are currently 5, 7, and 11 years old. How many years will it
be until they again have prime-numbered ages?
ANSWER: 6
2. What is the smallest 3-digit prime?
ANSWER: 101
3. Find a and b so that a + b = 25 and the difference of the square roots of a and b is 1.
ANSWER: 9,16
1:40
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 12 questions this round and a tiebreaker if necessary.
• You will have 30 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. If the midterm is worth 25% of the final grade in Jeff’s class and he scored an 60% on
the midterm, what is the highest final grade (expressed as a percentage) that Jeff can
achieve in the course?
ANSWER: 90%
2. 6 is the smallest perfect number. What is the next smallest perfect number?
ANSWER: 28
3. Iron Chef Michael Symon dices a block of cheese with side length 6 inches into smaller
cubes. How many smaller cubes are there if each has side length 1.5?
ANSWER: 64
4. Intuitively, when extending to projective space, where are the points added?
ANSWER: At infinity
5. What is the mathematical term for a geometric shape created by gluing the ends of a
cylinder together?
ANSWER: A torus. (Klein bottle is also acceptable)
6. Lori’s tank burns through three gallons of gas per mile but carries 600 gallons of gas.
Jim’s Ford Escort gets 20 miles per gallon but carries only 11 gallons. Which vehicle
can travel a greater distance on a full load of gas, and by how much?
ANSWER: the Ford Escort, by 20 miles (200 vs. 220)
1:40, page 2
7. How many numbers between 231 and 2301 are divisble by 23?
ANSWER: 90
8. Is the following statement about planar triangles true always, sometimes, or never?
Corresponding parts of congruent triangles are congruent?
ANSWER: Always
9. The moon is roughly 240,000 miles above the Earth’s surface. Assuming each fourteener
is exactly 14,000 feet, how many fourteeners would you need to stack end-to-end to
reach the moon’s orbit?
ANSWER: 18 (NOTE: We have 53 here in Colorado!)
10. A record player rotates three times per minute and has radius 5 inches. If an ant falls
asleep on the outer edge of the record player, how far will he travel (in terms of π) in
five minutes?
ANSWER: 150π (= 3 · 5 · 10π)
11. Jane is twice as old as Kelly. Lisa’s age is one third of Kelly’s age. In what year will
Jane’s age be twice Lisa’s age, if Kelly turned 3 this year on their shared birthday of
January 1st?
ANSWER: 2013
12. (READ SLOWLY!) A Boeing airplane carries 120 passengers and has 6 wheels. An
Airbus carries 160 passengers and has 10 wheels. If there are 680 passengers and 38
wheels on a tarmac, how many Boeings must there be?
ANSWER: 3 (2 Airbuses)
TIEBREAKER: What is 400 in base 20?
ANSWER: 100
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
1:40, page 3
Extra questions
1. Find the integer between 100 and 1000 that is both a square and a cube.
ANSWER: 729
2. Which real numbers are equal to their cubes?
ANSWER: -1, 0, 1
3. Neglecting the order of addition, in how many ways can 30 be written as the sum of
two primes.
ANSWER: 3
2:05
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 15 questions this round and a tiebreaker if necessary.
• You will have 45 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. What is the difference between maximum number of coins needed to make up $0.49
and the minimum number?
ANSWER: 42 (= 49 − 7)
2. Suppose a robot moves forward one foot, turns 90 degrees to the right, moves forward
two feet, turns 90 degrees to the right, moves forward three feet, turns 90 degrees to
the right, and so on. How far
√ is the robot from its starting point after 5 such moves?
√
ANSWER: 13 feet (= 22 + 32 )
3. Let A be the GCD of 148 and 222; let B be the LCM of 148 and 222; and let C be 148
times 222. Compute the quantity A times B and subtract off C. What do you get?
ANSWER: 0
4. In a race with 100 runners, how many different first place/second place/third place
outcomes are possible?
ANSWER: 970200 (= 100 · 99 · 98)
5. A spherical balloon’s diameter increases by 100%. By what percentage does its surface
area increase?
ANSWER: 300%
6. A 12-disc compilation of the greatest hits of Journey costs $90 in the store, plus 5% tax.
An infomercial offers it for three easy installments of just $29.99 plus $7.99 shipping
and handling. How much money will Jamie save if she buys the compilation in the
store?
ANSWER: $3.46
√
7. For the next two questions, let i = −1. What is eiπ ?
ANSWER: -1
2:05, page 2
8. What is the value of cos(x) + i sin(x) if x = π?
ANSWER: -1
9. What is the smallest number greater than 2 such that the remainder when divided by
15 is 1 and the remainder when divided by 14 is 1?
ANSWER: 211
10. Chris buys 105 M&Ms for his friends. He gives 1 to the first friend, 2 to the second, 3
to the third, and so on. How many friends will get at least one M&M?
ANSWER: 14
11. How many subsets does a set with 5 elements have, if we allow for subsets of sizes 0
through 5, inclusive?
ANSWER: 32
12. What is the equation of the line going through the point (π, π) and perpendicular to
the line y = −4.
ANSWER: x = π
13. How many numbers between 100 and 300 are divisible by 13?
ANSWER: 16
√
14. A cube with sidelength 2 3 is inscribed in a sphere of radius R. What is R?
ANSWER: 3
15. Expanding on the previous problem, what is the volume of the space between the
sphere and the cube, in terms of π?
ANSWER: 36π − 8 (= 34 π33 − 23 )
TIEBREAKER: Jimmy and Shirley celebrate New Year’s Day by checking their P.O.
boxes. From then on, Jimmy checks his box every third day and Shirley checks hers every
seventh day. Over the subsequent calendar year, on how many days will they both check
their respective P.O. boxes?
ANSWER: 17
2:05, page 3
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
2:05, page 4
Extra questions
1. Give all solutions to the equation sin(x) = x.
ANSWER: x = 0
2. What is the smallest positive integer that is both a square and a cube?
ANSWER: 1
3. What is the cube root of one million?
ANSWER: 100
4. How many integer solutions are there to the inequality 1 < x2 < 25?
ANSWER: 6
2:30
ANNOUNCEMENTS:
• This is a large/small school competition. (Which is it??)
• There are 15 questions this round and a tiebreaker if necessary.
• You will have 45 seconds to answer each question.
• Do not leave until we tell you where AND WHEN you are going next.
1. How many ways can you rearrange the letters of the word FORT?
ANSWER: 24 (= 4 ∗ 3 ∗ 2 ∗ 1)
2. How many ways can you rearrange the letters of the word COLLINS, assuming the
two L’s are identical?
ANSWER: 2520(= 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1/2)
3. Suppose a farmer starts with 10 pounds of compost and increases the weight of her
compost heap by 10% per year. How many pounds of compost will she have after four
years, rounded to the nearest tenth of a pound?
ANSWER: 14.6 pounds
4. How many seconds are there this month?
ANSWER: 2,592,000
5. Julius Caesar encoded messages by shifting each letter forward a certain number of
letters. For example, he often used a shift of 3 by changing A to D, B to E, and so on.
What shift was used to create the following encoded message? “MN RTR”
ANSWER: 5 (HI MOM)
(3!)!
6. Recall that k! is the product of the first k positive integers. What is 3! ?
ANSWER: 120
7. How many diagonals are there in an n-sided polygon?
n(n−3)
n(n−1)
ANSWER: Any of the following:
− n,
2 , (n choose 2) − n,
2
or 12 n2 − 23 n
8. Write this down. The production of a can of fruit cocktail requires 2 cups of fruit and
3 ounces of water. A can of fruit juice requires 1 cup of fruit and 8 ounces of water.
Fruit cocktail sells for $1 and fruit juice sells for $2. If you have 12 cups of fruit and
32 ounces of water available, what’s the most money you can make?
ANSWER: $9 (2 juices, 5 cocktails)
2:30, page 2
9. Express the difference of the binary numbers 1000 and 100 in binary.
ANSWER: 100
10. A knapsack is designed to hold up to 30 pounds. While robbing a hardware store,
you must choose between grabbing 3-pound drills worth $7 each and grabbing 9-pound
sledgehammers worth $22 each. If all of your loot must fit in your knapsack, what is
the most money you can hope to make from this robbery?
ANSWER: $73 (shame on you!)
11. Hexadecimal is the technical word for “base 16.” The digits in base 16 are 0 through
9 and A through F. What is the decimal value of the hexadecimal number BAD?
ANSWER: 2989 (= 13 + 10 ∗ 16 + 11 ∗ 162 )
12. In how many ways can a set of three distinct algebra books, three distinct geometry
books, and three distinct art history books be arranged on a shelf if the respective
groups must be kept together?
ANSWER: 1296 (6 within each group (63 ) times 6 ways to arrange the
groups)
13. Billy Bob has $1.65 in dimes and quarters. If he has three times as many dimes as
quarters, how many coins does he have?
ANSWER: 12
14. An ant starts its day at 8 a.m. located ten meters from a wall. Every minute, it walks
half of the remaining distance to the wall. What is the earliest time, rounded to the
nearest minute, at which the ant will be within 1 cm of the wall?
ANSWER: 8:10 a.m.
15. Some license plates consist of 3 numbers followed by 2 letters. How many license plates
of this type are there?
ANSWER: 676000
TIEBREAKER: There were two double elimination competitions today, each with 32
teams. Assuming the winner of each will finish with one losses, how many three-on-three
competitions happened today?
ANSWER: 126 (since that’s how many losses there were!)
2:30, page 3
ANNOUNCEMENTS:
• The final score of this round is:
• Tell them where they each compete next.
2:30, page 4
Extra questions
1. A positive integer is called perfectly balanced if adding and subtracting the same
positive integer results in two squares. What is the smallest perfectly balanced number?
ANSWER: 2
2. If a clock strikes the hour every hour, how many times will it strike during a 24-hour
period?
ANSWER: 156
3. If 2 cats can catch 2 mice in 2 days, how many mice cat 6 cats catch in 6 days?
ANSWER: 18
4. What is the area of the largest rectangle that can fit inside a circle of radius 1?
ANSWER: 2