Download Examples of Previous Qualifying Exams

Table of Contents Note: Qtest stands for qualifying test. Disclaimer from AMS's website: "Keep in mind that Who Wants To Be A Millionaire is an established
television show, which means the name and game are trademarks and copyrighted. The adaptation that
the AMS presents, Who Wants to Be a Mathematician, has been developed as a non-profit outreach
program for talented high school students. Teachers and others may use the sample questions and
format for their own teaching purposes only and may not distribute or profit from this game in any way." 1-2010-QualifyingTest-National
Qualifying Test for Who Wants to Be a Mathematician
Student Name:
Grade:
High School:
High School Address:
Contact Person:
State and zip:
Contact Person Phone:
Contact Person Email Address:
Test-taker acknowledges that, if selected as a contestant for the AMS's Who Wants to Be a
Mathematician contest, which selection belongs solely to the AMS based on the questions below and on
the attached test, he/she will abide by the rules of the contest and that the decisions of the AMS as to
prizes and eligibility thereto are solely at the discretion of the AMS.
What’s your favorite subject in school? ____________________________________________________
What’s your favorite non-school activity? __________________________________________________
__________________________________________________________________________________________
If you won the top prize, what would you do with the $5000? (in 30 words or less):
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
You don’t have to show your work on the test paper. Just write the final answer. No
calculators. You have ten minutes (for the problems on the next page). Good luck!
1-2010-Qualifying-Test-National
1.
What is the tens digit (the digit second from the right) of 112009? _________ .
2.
What is the radius of the circle with equation x − 4 x = 1 − y − 6 y ? ____________
3.
How many vertices does a regular icosahedron have? _________
4.
1⎞
⎛
cos ⎜ 2sin −1 ⎟ = ______________
5⎠
⎝
2
2
5.
Put the following mathematicians in order according to their year of birth, starting with the first
born: Galois, Gauss, Hilbert, Newton. ______________________________________________
6.
Find a fourth-degree polynomial with real coefficients that has i and 2 − i as roots. (Do not leave
your answer in factored form.)
Ans: ____________________________
7.
A triangle, M, is formed from
ABC by constructing segments that connect the midpoints of
the three sides. What is the ratio of the area of M to the area of
8.
ABC? ________________
How many odd numbers are in the 17th row of Pascal’s triangle (where the 0th row is 1 and the
1st row is 1 1)? ______________
9. A Pythagorean triple (a, b, c) consists of three positive integers such that a2+ b2 = c2. Write all
Pythagorean triples that contain the number 37. (Consider triples in which a and b are
interchanged to be equal, that is (3, 4, 5) and (4, 3, 5) are regarded as one triple.)
Ans: ____________________________________________________
10. A googol (in base ten) is 1 followed by one hundred zeros. Within ten, how many digits are there
in a googol written in base five? ________________
Thank you for participating.
Return completed test(s)—so that they arrive by Oct. 20, 2009—to Mike Breen; c/o American
Mathematical Society; 201 Charles St.; Providence, RI 02904.
1-2010-qual-test-answers
1.
What is the tens digit (the digit second from the right) of 112009?
2.
3.
What is the radius of the circle with equation x − 4 x = 1 − y − 6 y ?
How many vertices does a regular icosahedron have?
12
4.
1⎞
⎛
cos ⎜ 2sin −1 ⎟ =
5⎠
⎝
2
9
2
14
23
25
5.
Put the following mathematicians in order according to their year of birth, starting with the first
born: Galois, Gauss, Hilbert, Newton.
Newton, Gauss, Galois, Hilbert
6.
Find a fourth-degree polynomial with real coefficients that has i and 2 − i as roots. (Do not leave
your answer in factored form.)
(many answers possible, one is
7.
A triangle, M, is formed from
x 4 − 4 x3 + 6 x 2 − 4 x + 5 )
ABC by constructing segments that connect the midpoints of
the three sides. What is the ratio of the area of M to the area of
ABC? 1/4
8. How many odd numbers are in the 17th row of Pascal’s triangle (where the 0th row is 1 and the
1st row is 1 1)?
4
9. A Pythagorean triple (a, b, c) consists of three positive integers such that a2+ b2 = c2. Write all
Pythagorean triples that contain the number 37. (Consider triples in which a and b are
interchanged to be equal, that is (3, 4, 5) and (4, 3, 5) are regarded as one triple.)
(12,35,37), (37,684,685)
10. A googol (in base ten) is 1 followed by one hundred zeros. Within ten, how many digits are there
in a googol written in base five?
Accept anything between 134 and 154
Thank you for participating.
2-2011-qtest-national
Qualifying Test for Who Wants to Be a Mathematician
Student Name:
Grade:
High School:
High School Address:
Teacher:
State and zip:
Teacher Phone:
Teacher Email Address:
Test-taker acknowledges that, if selected as a contestant for the AMS's Who Wants to Be a
Mathematician contest, which selection belongs solely to the AMS based on the questions below and on
the attached test, he/she will abide by the rules of the contest and that the decisions of the AMS as to
prizes and eligibility thereto are solely at the discretion of the AMS.
What’s your favorite subject in school? ____________________________________________________
What’s your favorite non-school activity? __________________________________________________
__________________________________________________________________________________________
If you won the top prize, what would you do with the $5000? (in 30 words or less)
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
Fill in the blank (many correct answers possible):
“How, I, wish, I, could, enumerate, ___________________”
You don’t have to show your work on the test paper. Just write the final answer. No
calculators. You have ten minutes (for the problems on the next page). Good luck!
Return completed test(s)—so that they arrive by Oct. 22—to Mike Breen; c/o American
Mathematical Society; 201 Charles St.; Providence, RI 02904.
2-2011-qtest-national
1.
In how many points does the line x + y = 2 intersect the circle x + y = 2 ? ____________
2.
What is the largest four-digit prime number less than 2010? _________ .
3.
How many five-letter “words” (any strings of five characters from the English alphabet) strictly
alternate between vowels and consonants (one example is “mimes”)? _________________ (You
may leave your answer in factored form; consider the letter y to be a consonant.)
4.
What is the highest power of 2 that divides 100! (without remainder)? ______________
5.
Put the following mathematicians in order according to their year of birth, starting with the first
born: A. Emmy Noether, B. Leonhard Euler, C. John Nash, D. Archimedes. (use the indicated
letters)
________________________________________________________
6.
Solve for x: 2 x − 3 x − 8 x = 3 .
7.
How many vertical asymptotes does the graph of y = tan(sec x ) have in the interval ⎡⎣ 0,
2
3
2
2
π
2
)?
______________________
8.
Put the following events in order from the least likely to the most likely (use the indicated letters):
E: Rolling four fair six-sided dice and getting a sum of 5.
F: 10 fair two-sided coins landing “heads”
G: Choosing a palindrome (a number that reads the same backwards and forwards) at random from
among all seven-digit numbers
9. The area of a circle inscribed in an equilateral triangle is 1 sq. ft. What is the perimeter of the
triangle? ________________
10. Which of the following is the negation of the statement “For every x there is a y such that if x has
property P then y has property Q”? (Circle the letter of your answer.)
A. There is an x such that for every y, x has property P and y does not have property Q
B. For every x there is no y such that x has property P and y has property Q
C. There is an x such that for every y, if x does not have property P then y does not have property Q
Thank you for participating.
2-2011-qtest-national-answers
1.
In how many points does the line x + y = 2 intersect the circle x + y = 2 ? Ans: One
2.
What is the largest four-digit prime number less than 2010? Ans: 2003 .
3.
How many five-letter “words” (any strings of five characters from the English alphabet) strictly
alternate between vowels and consonants (one example is “mimes”)? (You may leave your
answer in factored form; consider the letter y to be a consonant.) Ans: 26*1052 (other forms
possible)
4.
What is the highest power of 2 that divides 100! (without remainder)? Ans: 97 (or 297)
5.
Put the following mathematicians in order according to their year of birth, starting with the first
born: A. Emmy Noether, B. Leonhard Euler, C. John Nash, D. Archimedes. (use the indicated
letters)
Ans: D B A C
6.
Solve for x: 2 x − 3 x − 8 x = 3 . Ans: -1, -1/2, 3
7.
How many vertical asymptotes does the graph of y = tan(sec x ) have in the interval ⎡⎣ 0,
2
3
2
2
π
2
)?
Ans: An infinite number
8.
Put the following events in order from the least likely to the most likely (use the indicated letters):
E: Rolling four fair six-sided dice and getting a sum of 5.
F: 10 fair two-sided coins landing “heads”
G: Choosing a palindrome (a number that reads the same backwards and forwards) at random from
among all seven-digit numbers
Ans: F G E
9. The area of a circle inscribed in an equilateral triangle is 1 sq. ft. What is the perimeter of the
triangle? Ans:
6 3
π
(other forms possible)
10. Which of the following is the negation of the statement “For every x there is a y such that if x has
property P then y has property Q”? (Circle the letter of your answer.)
A. There is an x such that for every y, x has property P and y does not have property Q
B. For every x there is no y such that x has property P and y has property Q
C. There is an x such that for every y, if x does not have property P then y does not have property Q
Ans: A
Thank you for participating.
3-2012-qtest-National
sin −1 ( sin10 )
1.
Find (the principal value of) :
(10 radians, not degrees)
2.
What is the highest power of 5 that divides 2011! ?
3.
How many real solutions are there to the equation
4.
True or False ? (circle one) No path that begins and ends at A traverses each segment exactly
once. (The two diagonals each count as one segment; ignore their point of intersection.)
x − 1 =4 x ?
A
B
D
C
5.
Use digits a, b, and c to form a three-digit number abc. How many such numbers between 100
and 200 are prime and have the property that ab, ac, and bc (each considered as two-digit
numbers) are themselves all prime?
6.
Which of the following Greek mathematicians was known as “Beta”? (circle one)
A. Archimedes
B. Eratosthenes
C. Euclid D.Pythagoras
7.
Put the following events in order from the least likely to the most likely (use the indicated letters):
E: Tossing six fair coins and getting exactly three heads
F: Rolling two fair six-sided dice and getting a sum of 6 or 7
G: Choosing a letter from the English alphabet (26 letters) at random and getting a letter that
either immediately precedes or immediately follows a vowel. (Here we are not counting Y
as a vowel and we assume that the alphabet ends at Z—it doesn’t wrap back to A.)
8.
How many non-real solutions are there to the equation 12 x8 − 3 x 4 − 15 =
0?
9.
A unit cube (each side has length 1) is inscribed in a sphere. What is the surface area of the
sphere?
10.
How many positive numbers x satisfy the equation ?
x x −1 =10 ?
Thank you for participating.
3-2012-qtest-national-answers
Answers in red boxes.
1.
Find (the principal value of) sin −1 ( sin10 ) (10 radians, not degrees)
2.
What is the highest power of 5 that divides 2011! ?
3.
How many real solutions are there to the equation
4.
True or False (circle one)? No path that begins and ends at A traverses each segment exactly
once. (The two diagonals each count as one segment; ignore their point of intersection.)
3π − 10
501
x −1 = 4 x ?
1
A
B
D
C
5.
Use digits a, b, and c to form a three-digit number abc. How many such numbers between 100
and 200 are prime and have the property that ab, ac, and bc (each considered as two-digit
numbers) are themselves all prime?
6
6.
Which of the following Greek mathematicians was known as “Beta”? (circle one)
A. Archimedes
C. Euclid
D. Pythagoras
B. Eratosthenes
7.
Put the following events in order from the least likely to the most likely (use the indicated letters):
E: Tossing six fair coins and getting exactly three heads (5/16)
F: Rolling two fair six-sided dice and getting a sum of 6 or 7 (11/36)
G: Choosing a letter from the English alphabet (26 letters) at random and getting a letter that
either immediately precedes or immediately follows a vowel. (Here we are not counting Y
as a vowel and we assume that the alphabet ends at Z—it doesn’t wrap back to A.) (9/26)
(above values not required)
FEG
8.
How many non-real solutions are there to the equation 12 x8 − 3 x 4 − 15 = 0 ?
9.
A unit cube (each side has length 1) is inscribed in a sphere. What is the surface area of the
sphere?
6
3π
10.
How many positive numbers x satisfy the equation ?
x x −1 =10 ?
2
Thank you for participating.
4-2013-qtest
𝑥 2 = 2𝑥 ?
1.
How many real solutions are there to
_________________
2.
What is the largest difference between three-digit positive integers (those between 100 and 999,
inclusive) that are reversals of each other (abc – cba) ?
________________
3.
What is the minimum value of sec(𝜃) ∙ csc(𝜃) on the interval (0, 𝜋⁄2) ?
_______________
4.
What is the area of an n-sided polygon with perimeter p that is circumscribed about a circle of
radius r?
_______________
5.
Put the following three numbers in increasing order (you can use the letters a, b, and c):
a. 1002012 b. 1006! ∙ 1006! c. 2012!
__________________
6.
The complex number i is one root (zero) of the polynomial 2x4 + x3 – 4x2 + x – 6. Find all
other roots.
__________________
7.
Two fair, six-sided dice are tossed. What is the probability that the sum (of the spots showing) is
a prime number?
______________________
8.
Put these mathematicians in order according to where they were born, starting at the International
Date Line, which is just west of Hawai'i, and moving east. (Use the letters for your answer.)
a. Srinivasa Ramanujan b. John Nash c. Emmy Noether d. William Rowan Hamilton
______________________
9.
How many times from noon until midnight are the continuously moving hour and minute hands
of an analog clock at an angle of 180 degrees?
______________________
10.
A square of area 1 is inscribed in a larger square (as shown) so that the length of AB is three
times the length of BC. What is the area of the larger square?
______________________
A
B
C
Thank you for participating.
4-2013-qtest-answers
𝑥 2 = 2𝑥 ?
1.
How many real solutions are there to
________3_________
2.
What is the largest difference between three-digit positive integers (those between 100 and 999,
inclusive) that are reversals of each other (abc – cba) ?
______792_____
3.
What is the minimum value of sec(𝜃) ∙ csc(𝜃) on the interval (0, 𝜋⁄2) ?
____2_____
4.
What is the area of an n-sided regular polygon with perimeter p that is circumscribed about a
circle of radius r?
____pr/2_____
5.
Put the following three numbers in increasing order (you can use the letters a, b, and c):
a. 1002012 b. 1006! ∙ 1006! c. 2012!
_____abc_____
6.
The complex number i is one root (zero) of the polynomial 2x4 + x3 – 4x2 + x – 6. Find all
other roots.
____ -i, -2, 3/2___
7.
Two fair, six-sided dice are tossed. What is the probability that the sum (of the spots showing) is
a prime number? [Note: we meant the spots showing on top, the usual dice method.]
______5/12________
8.
Put these mathematicians in order according to where they were born, starting at the International
Date Line, which is just west of Hawai'i, and moving east. (Use the letters for your answer.)
a. Srinivasa Ramanujan b. John Nash c. Emmy Noether d. William Rowan Hamilton
_____bdca_____
9.
How many times from noon until midnight are the continuously moving hour and minute hands
of an analog clock at an angle of 180 degrees?
_______11______
10.
A square of area 1 is inscribed in a larger square (as shown) so that the length of AB is three
times the length of BC. What is the area of the larger square?
____8/5______
A
B
C
Thank you for participating.
5a-2014-qtest-roundone
Round One Qualifying Test for Who Wants to Be a Mathematician
Student Name:
Grade:
High School:
HS Address (incl. town/st./zip):
Contact Person:
Contact Person Phone:
Contact Person Email Address:
Test-taker acknowledges that, if selected as a contestant for the AMS's WWtBaM contest, which selection belongs solely to the AMS, he/she will abide by
the rules of the contest and that the decisions of the AMS as to prizes and eligibility thereto are solely at the discretion of the AMS.
You don’t have to show your work on this paper. Just write the final answer. No calculators.
You have 15 minutes. Good luck!
2.
Solve for 𝑥𝑥: 2𝑥𝑥 3 + 9𝑥𝑥 2 = 35𝑥𝑥. _________
3.
How many (positive integer) factors does 1000 have (including 1 and itself)?
1.
Find sec2(𝜃𝜃) if tan(𝜃𝜃) = −√2 . ______________________
_________
5.
What is the coefficient of the 𝑥𝑥 2 𝑦𝑦 3 term in the expansion of (2𝑥𝑥 − 𝑦𝑦)5 ? _________________
6.
(Circle your answer.) The 19th century mathematician Niels Abel was born in what is now
4.
An equilateral triangle is inscribed in a circle of radius 1. What is the area of the region inside the
circle but outside the triangle? __________
a. England
b. France
c. Norway
d. Scotland
7.
Two cards are drawn without replacement from a standard deck of 52 cards. What is the
probability that both are of the same suit?
____________________
8.
(Circle your answer.) The hypotenuses of two right triangles have the same length. The ratio of
the lengths of the legs of the first triangle is 4:3 while the ratio of the lengths of the legs of the
second triangle is 16:9. Which of the following is true about the length of the smallest side of the
first triangle divided by the length of the smallest side of the second triangle? It's
a. between ½ and ¾
b. between ¾ and 1
c. between 1 and 5/4
d. greater than 5/4
9.
Suppose a and b are positive integers greater than 1. If log 𝑎𝑎 √𝑏𝑏 = 𝑠𝑠 then what is log 𝑏𝑏 (𝑎𝑎2 )?
______________
10.
(Circle your answer.) What is the largest number that cannot be written in the form
6a + 9b + 20c, where a, b, and c are non-negative integers?
a. 22
b. 23
c. 28
d. 37
e. 43
Return completed test(s) to Mike Breen (email: [email protected]; fax: 401-331-3842; or
mail: c/o American Mathematical Society; 201 Charles St.; Providence, RI 02904)
Thank you for participating.
5a-2014-qtest-roundone-answers
WWTAM 2014 Round One National Contest Qualifying Test
(answers in red)
1. Solve for 𝑥: 2𝑥 3 + 9𝑥 2 = 35𝑥 .
0, 5/2, −7
2. Find sec2(𝜃) if tan(𝜃) = −√2 .
3
3. How many (positive integer) factors does 1000 have (including 1 and itself)?
16
4. What is the coefficient of the 𝑥 2 𝑦 3 term in the expansion of (2𝑥 − 𝑦)5 ?
−40
5. An equilateral triangle is inscribed in a circle of radius 1. What is the area of the region inside the
circle but outside the triangle?
𝜋−
3√3
4
6. The 19th century mathematician Niels Abel was born in what is now
a. England
b. France
c. Norway
d. Scotland
7. Two cards are drawn without replacement from a standard deck of 52 cards. What is the
probability that both are of the same suit?
4/17 (or any fraction equal to 4/17)
8. The hypotenuses of two right triangles have the same length. The ratio of the lengths of the legs
of the first triangle is 4:3 while the ratio of the lengths of the legs of the second triangle is 16:9.
Which of the following is true about the length of the smallest side of the first triangle divided
5a-2014-qtest-roundone-answers
by the length of the smallest side of the second triangle? (Again, just enter the letter of your
answer.)
a. It's between ½ and ¾
b. It's between ¾ and 1
c. It′s between 1 and 5/4
d. It's greater than 5/4
9. Suppose a and b are positive integers greater than 1. If log 𝑎 √𝑏 = 𝑠 then what is log 𝑏 (𝑎2 )?
1/s
10. What is the largest number that cannot be written in the form 6a + 9b + 20c, where a, b, and c
are non-negative integers? (Again, just enter the letter of your answer.)
a. 22
b. 23
c. 28
d. 37
e. 43
5b-2014-qtest-roundtwo
2014 Qualifying Test Round Two
1. How many real solutions are there to the equation cos 𝑥𝑥 = ln 𝑥𝑥 ?
1
𝑥𝑥
2. What is the distance between the foci of the hyperbola y = ?
3. The general solution to the cubic equation was published in 1545 in Ars Magna, which was authored by which
mathematician?
a. Bombelli
b. Cardano
c. Descartes
d. Pascal
4. How many real solutions are there in the interval [0,2𝜋𝜋] to the equation 2 sin2 𝜃𝜃 + 2 sin 𝜃𝜃 = 1?
5. Which of the following is the negation of the statement "P implies (Q and R)"?
a. P and (Q implies not R)
b. P and not (Q or R)
c. (Not P) and (Q implies not R)
d. (Not P) and (Q implies R)
6. In a group of 20 randomly chosen people, what is the probability that exactly one person has the same birthday as
you? (Assume 365 days in a year, and that birthdays are distributed uniformly.)
7. An ice cream cone shape is formed by gluing the base of a cone with radius 2 cm and height 3cm to the base of a
hemisphere with the same radius. What is the total volume of the ice cream cone (in cubic cm)?
8. A set of n + 1 numbers is chosen from the set {1,2,3,…,2n} where n > 1 and the greatest common factor (greatest
common divisor) of each pair of numbers from the chosen set is determined. What is the minimum value of these
greatest common factors (greatest common divisors)?
a. 1
b. 2
𝑛𝑛
𝑛𝑛
c. � � (the greatest integer part of )
2
2
d. It depends on n and the chosen set.
9. How many integers larger than 500 (in base ten) consist of only distinct even digits?
10. The rhombus (a quadrilateral with four congruent sides) pictured has sides of length 1 and a diagonal of length 1. What
is the radius of the inscribed circle?
5b-2014-qtest-roundtwo-answers
2014 Qualifying Test Round Two Part Two [Answers in brackets in red]
1. How many real solutions are there to the equation cos 𝑥 = ln 𝑥 ? [1]
1
𝑥
2. What is the distance between the foci of the hyperbola y = ? [4]
3. The general solution to the cubic equation was published in 1545 in Ars Magna, which was authored by
which mathematician?
a. Bombelli
b. Cardano
c. Descartes
d. Pascal
4. How many real solutions are there in the interval [0,2𝜋] to the equation 2 sin2 𝜃 + 2 sin 𝜃 = 1? [2]
5. Which of the following is the negation of the statement "P implies (Q and R)"?
a. P and (Q implies not R)
b. P and not (Q or R)
c. (Not P) and (Q implies not R)
d. (Not P) and (Q implies R)
6. In a group of 20 randomly chosen people, what is the probability that exactly one person has the same
birthday as you? (Assume 365 days in a year, and that birthdays are distributed uniformly.)
[20*364^19/(365^20)] (also assume you're not in the group)
7. An ice cream cone shape is formed by gluing the base of a cone with radius 2 cm and height 3cm to the
base of a hemisphere with the same radius. What is the total volume of the ice cream cone (in cubic
cm)? [28pi/3]
8. A set of n + 1 numbers is chosen from the set {1,2,3,…,2n} where n > 1 and the greatest common factor
(greatest common divisor) of each pair of numbers from the chosen set is determined. What is the
minimum value of these greatest common factors (greatest common divisors)?
a. 1
b. 2
𝑛
𝑛
c. � � (the greatest integer part of )
2
2
d. It depends on n and the chosen set.
9. How many integers larger than 500 (in base ten) consist of only distinct even digits? [216]
10. The rhombus (a quadrilateral with four congruent sides) pictured has sides of length 1 and a diagonal of
length 1. What is the radius of the inscribed circle? [sqrt(3)/4]
6a-2015-qtest-roundone
2015 Qualifying Test Round One
1.
Find the slope of the line whose equation is 2𝑦 − 3𝑥 = 5.
2.
What is the fourth digit to the right of the decimal point in the decimal expansion of 𝜋 ? _____
3.
The x-coordinate of the point where the graph of xy = 8 and y = x2 intersect is
4.
5.
_________
_________
sin �2 cos −1�4�5�� = _________________
Which platonic solid has sides shaped like pentagons?
a. dodecahedron
b. icosahedron
c. octahedron
d. tetrahedron
6.
What is the smallest degree possible for a polynomial with integer coefficients that has
5/2 and 3 + 2i as roots ?
_____________________
7.
The million-dollar Millennium Problem that has been solved was named for
a. David Hilbert
b. William Hodge
c. Henri Poincaré
d. Bernhard Riemann
8.
Find the sum of the following finite sequence, the alternating sum of the first 2,014 positive
integers: 1 – 2 + 3 – 4 + …+ 2013 – 2014 .
____________________
9.
How many odd numbers between 100 and 999 have three distinct digits? ______________
10.
What is the greatest common factor (greatest common divisor) of the 66th term and the 300th term
of the Fibonacci sequence (1, 1, 2, 3, 5, …) ?
_____________
Return completed test(s) to Mike Breen (email: [email protected]; fax: 401-331-3842; or
mail: c/o American Mathematical Society; 201 Charles St.; Providence, RI 02904)
Thank you for participating.
6a-2015-qtest-roundone-answers
Round One Qualifying Test
2015 Who Wants to Be a Mathematician
Test-taker acknowledges that, if selected as a contestant for the AMS's WWtBaM contest, which
selection belongs solely to the AMS, he/she will abide by the rules of the contest and that the decisions
of the AMS as to prizes and eligibility thereto are solely at the discretion of the AMS.
You don’t have to show your work on this paper. Just write the final answer. No calculators.
You have 15 minutes. Good luck!
2.
Find the slope of the line whose equation is 2𝑦 − 3𝑥 = 5.
3.
The x-coordinate of the point where the graph of xy = 8 and y = x2 intersect is
1.
4.
5.
___3/2______
What is the fourth digit to the right of the decimal point in the decimal expansion of 𝜋 ? __5___
___2_____
sin �2 cos −1�4�5�� = ______24/25___________
Which platonic solid has sides shaped like pentagons?
a. dodecahedron
b. icosahedron
c. octahedron
d. tetrahedron
6.
What is the smallest degree possible for a polynomial with integer coefficients that has
5/2 and 3 + 2i as roots ?
________3_____________
7.
The million-dollar Millennium Problem that has been solved was named for
a. David Hilbert
b. William Hodge
c. Henri Poincaré
d. Bernhard Riemann
8.
Find the sum of the following finite sequence, the alternating sum of the first 2,014 integers:
1 – 2 + 3 – 4 + …+ 2013 – 2014?
_______-1007_____________
9.
How many odd numbers between 100 and 999 have three distinct digits? ____320__________
10.
Let Fn denote the nth term of the Fibonacci sequence 1, 1, 2, 3, 5, … . What is the greatest
common factor (greatest common divisor) of F66 and F300?
_____8________
Thank you for participating.
6b-2015-qtest-round-two
2015 Qualifying Test Round Two
1.
What is the ones (units) digit of 257,885,161 − 1 ?
2.
A fair coin is tossed three times (and whether it lands heads or tails each time is noted). The same
coin is then tossed three more times. What is the probability that the first sequence of tosses
matches the second sequence?
_____
3.
Which of the following is largest?
𝜋𝜋
𝜋𝜋
a. cos �cos 6 � b. cos �sin 6 �
_________
𝜋𝜋
c. sin �cos 6 �
𝜋𝜋
d. sin �sin 6 �
(circle one)
4.
What is the coefficient of the x2 term in (3𝑥𝑥 + 2)5 − (2𝑥𝑥 + 3)3 ?
5.
Three of the vertices of a square in R3 are (1, 2, 3), (10, 14, 23), and (22, 30, 8). What is the sum
of the coordinates of the fourth vertex?
_________________
6.
An n-term arithmetic sequence (n > 1) with common difference d starts with 4 and ends with
2014. Write d as a function of n.
d = _____________________
7.
A right triangle has legs of length 7 and 24. What is the length of the altitude that extends from
the vertex at the right angle to the hypotenuse?
____________________
8.
A set of numbers in which the leading digit, D, occurs with probability log10 (1 + 1/𝐷𝐷) obeys a
law named after (circle one):
a. Frank Benford
b. Pafnuty Chebyshev
c. Andrei Markov
_________________
d. John von Neumann
9.
How many quadratic functions with leading coefficient 2 have two distinct integer roots and a
graph with a y-intercept of 72?
______________
10.
For which value of n below is a regular n-sided polygon not constructible?
a. 255
b. 256
c. 257
d. 258
(circle one)
Thank you for participating.
6b-2015-qtest-roundtwo-answers
Round Two Qualifying Test
2015 National Who Wants to Be a Mathematician
1.
What is the ones (units) digit of 257,885,161 − 1 ?
2.
A fair coin is tossed three times (and whether it lands heads or tails each time is noted). The same
coin is then tossed three more times. What is the probability that the first sequence of tosses
matches the second sequence?
__1/8___
3.
Which of the following is largest?
𝜋𝜋
𝜋𝜋
a. cos �cos 6 � b. cos �sin 6 �
____1_____
𝜋𝜋
c. sin �cos 6 �
𝜋𝜋
d. sin �sin 6 �
(circle one)
4.
What is the coefficient of the x2 term in (3𝑥𝑥 + 2)5 − (2𝑥𝑥 + 3)3 ?
5.
Three of the vertices of a square in R3 are (1, 2, 3), (10, 14, 23), and (22, 30, 8). What is the sum
of the coordinates of the fourth vertex?
_____19____________
6.
An n-term arithmetic sequence (n > 1) with common difference d starts with 4 and ends with
2014. Write d as a function of n.
d = ___2010/(n – 1)__________________
7.
A right triangle has legs of length 7 and 24. What is the length of the altitude that extends from
the vertex at the right angle to the hypotenuse?
_______168/25_____________
8.
A set of numbers in which the leading digit, D, occurs with probability log10 (1 + 1/𝐷𝐷) obeys a
law named after (circle one):
a. Frank Benford
b. Pafnuty Chebyshev
c. Andrei Markov
_____684____________
d. John von Neumann
9.
How many quadratic functions with leading coefficient 2 have two distinct integer roots and a
graph with a y-intercept of 72?
______8________
10.
For which value of n below is a regular n-sided polygon not constructible?
a. 255
b. 256
c. 257
d. 258
(circle one)
Thank you for participating.
7a-2016-qtest-roundone
Round One Qualifying Test
2016 National Who Wants to Be a Mathematician
1.
Solve for 𝑥𝑥: 2𝑥𝑥 2 − 𝑥𝑥 = 15 .
2.
What is the ones digit of 20172015 ?
3.
How many solutions are there in [0,2𝜋𝜋] to the equation sin 𝑥𝑥 = cos 𝑥𝑥 ?
4.
George Boole, who developed the logic upon which computers operate, was born in which of the
following countries?
a. Austria
b. England
c. France
d. Germany
5.
How many subsets (including the set itself) of a four-element set have at least two elements?
_________
________
______
_______________
6.
What is the area of the quadrilateral formed by gluing together a 3-4-5 right triangle and a 5-1213 right triangle along their common side of length 5 ?
_______________
7.
How many real solutions are there to the equation ln(𝑥𝑥 2 + 𝑥𝑥) = ln(𝑥𝑥 2 ) + ln(𝑥𝑥) ? ________
8.
A prism has an equilateral triangle as its base and top and three rectangles for its lateral sides. A
sphere of radius 1 fits inside the prism, touching all five of its faces. What is the volume of the
prism?
_________________
9.
Bob and Jane have three children. Given that one child is their daughter Mary, what is the
probability that Bob and Jane have at least two daughters?
______________
[Note: This question was ambiguous, so we accepted two right answers.]
10.
What integer is closest to the square root of the product of the largest two-digit prime number
and the smallest three-digit prime number?
_____________
Return completed test(s) to Mike Breen (email: paoffice at ams dot org; fax: 401-331-3842; or
mail: c/o American Mathematical Society; 201 Charles St.; Providence, RI 02904)
Thank you for participating.
7a-2016-qtest-roundone-answers
Round One Qualifying Test
2016 National Who Wants to Be a Mathematician
1.
Solve for 𝑥𝑥: 2𝑥𝑥 2 − 𝑥𝑥 = 15 .
2.
What is the ones digit of 20172015 ?
𝟑𝟑
3.
How many solutions are there in [0,2𝜋𝜋] to the equation sin 𝑥𝑥 = cos 𝑥𝑥 ?
𝟐𝟐
4.
George Boole, who developed the logic upon which computers operate, was born in which of the
following countries?
a. Austria
𝐛𝐛. 𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄𝐄
c. France
d. Germany
5.
How many subsets (including the set itself) of a four-element set have at least two elements?
−𝟓𝟓/𝟐𝟐, 𝟑𝟑
𝟏𝟏𝟏𝟏
6.
What is the area of the quadrilateral formed by gluing together a 3-4-5 right triangle and a 5-1213 right triangle along their common side of length 5 ?
𝟑𝟑𝟑𝟑
7.
How many real solutions are there to the equation ln(𝑥𝑥 2 + 𝑥𝑥) = ln(𝑥𝑥 2 ) + ln(𝑥𝑥) ?
8.
A prism has an equilateral triangle as its base and top and three rectangles for its lateral sides. A
sphere of radius 1 fits inside the prism, touching all five of its faces. What is the volume of the
prism?
𝟔𝟔√𝟑𝟑
9.
Bob and Jane have three children. Given that one child is their daughter Mary, what is the
probability that Bob and Jane have at least two daughters?
𝟒𝟒/𝟕𝟕 (we also accepted ¾)
10.
What integer is closest to the square root of the product of the largest two-digit prime number
and the smallest three-digit prime number?
𝟗𝟗𝟗𝟗
𝟏𝟏
Thank you for participating.
7b-2016-qtest-roundtwo
Round Two Qualifying Test
2016 National Who Wants to Be a Mathematician
1.
An integer between 2 and 7 inclusive is chosen at random. If a pair of fair dice is rolled, which
sum (of the top numbers on the two dice) has the same probability of appearing as the randomly
chosen integer?
_________
2.
How many zeros are at the end (rightmost digits) of 2015! ?
3.
In a geometric series ∑∞
𝑛𝑛=1 𝑎𝑎𝑛𝑛 , 𝑎𝑎2 = 54 and 𝑎𝑎5 = 2 . What is the sum of the series? ________
4.
A piece of fruit is a perfect sphere of radius r and has a hard spherical seed at its center of radius 1.
If the seed is removed, the volume of the remaining fruit is 7 times the volume of the seed. Find r.
_________________
5.
Which of the following mathematicians was one of the inventors of game theory?
a. John von Neumann b. Kurt Gödel c. George Pólya d. Paul Erdős (circle one)
6.
How many even six-digit numbers use every one of the six digits 0,1,2,3,4,5 ?
7.
Which of the following definitions of the binary operation * on the nonzero rational numbers
defines an associative operation? (“max” below denotes the maximum, if m = n, choose m)
a. m * n = m – n
b. m * n = 2m + 4n
c. m * n = mn
____________________
d. m * n = max{m,n}
____________
(circle one)
8.
Let 𝑎𝑎0 = 10 and for each positive integer n, let 𝑎𝑎𝑛𝑛 = 100𝑎𝑎𝑛𝑛−1 + (𝑛𝑛 + 10). For how many n,
________________
0 ≤ 𝑛𝑛 ≤ 100, is it true that 𝑎𝑎𝑛𝑛 is a multiple of 3?
9.
Suppose �9 + 4√5 = 𝑎𝑎 + 𝑏𝑏√5 where a and b are integers. Find a + b.
_____________
10. How many of the elements of the set {11, 111, 1111, … , 1111111111} (all numbers are
base 10; first number is eleven, then one hundred eleven, etc. 10 1’s in the last number) are prime?
_________________
Thank you for participating and congratulations on qualifying for Round Two.
7b-2016-qtest-roundtwo-answers
Round Two Qualifying Test
2016 National Who Wants to Be a Mathematician
1.
An integer between 2 and 7 inclusive is chosen at random. If a pair of fair dice is rolled, which
sum (of the top numbers on the two dice) has the same probability of appearing as the randomly
chosen integer?
_________
7
2.
How many zeros are at the end (rightmost digits) of 2015! ?
____________________
502
3.
In a geometric series ∑∞
𝑛𝑛=1 𝑎𝑎𝑛𝑛 , 𝑎𝑎2 = 54 and 𝑎𝑎5 = 2 . What is the sum of the series? ________
243
4.
A piece of fruit is a perfect sphere of radius r and has a hard spherical seed at its center of radius 1.
If the seed is removed, the volume of the remaining fruit is 7 times the volume of the seed. Find r.
_________________
2
5.
Which of the following mathematicians was one of the inventors of game theory?
a. John von Neumann b. Kurt Gödel c. George Pólya d. Paul Erdős (circle one)
6.
How many even six-digit numbers use every one of the six digits 0,1,2,3,4,5 ?
____________
312
7.
Which of the following definitions of the binary operation * on the nonzero rational numbers
defines an associative operation? (“max” below denotes the maximum, if m = n, choose m)
a. m * n = m – n
8.
b. m * n = 2m + 4n
c. m * n = mn
d. m * n = max{m,n} (circle one)
Let 𝑎𝑎0 = 10 and for each positive integer n, let 𝑎𝑎𝑛𝑛 = 100𝑎𝑎𝑛𝑛−1 + (𝑛𝑛 + 10). For how many n,
0 ≤ 𝑛𝑛 ≤ 100, is it true that 𝑎𝑎𝑛𝑛 is a multiple of 3?
________________
67
9.
Suppose �9 + 4√5 = 𝑎𝑎 + 𝑏𝑏√5 where a and b are integers. Find a + b.
_____________
3
10. How many of the elements of the set {11, 111, 1111, … , 1111111111} (base 10; starting with
eleven, then one hundred eleven, etc. 10 1’s in the last number) are prime? _________________
1
Thank you for participating.
8a-sample-national-semifinals
Semifinal
8a-sample-national-semifinals
Question #1—100 Points
30
11,111,011,1112 =
0
A. 1829
0
B. 1891
0
C. 2015
0
D. 4063
Response
8a-sample-national-semifinals
Question #2—200 Points
0
40
Four points on a circle of diameter 2 units are
connected to make a quadrilateral. What is the
largest possible area (in square units) of a
quadrilateral determined this way?
A.
0
B.
0
C.
0
D.
1
2
𝜋𝜋
4
Response
8a-sample-national-semifinals
Question #3—300 Points
40
The following lemma is attributed to which of the following mathematicians?
Lemma: Consider a big triangle whose corners are labelled 1, 2, and 3. Suppose it is
subdivided into smaller triangles (that meet nicely along edges) and such that
1. Edge vertices are labelled with one of the two labels from the corners
that span that edge
2. Vertices inside the triangle are arbitrarily labelled either 1, 2, or 3.
Then there exists a smaller triangle whose vertices have all labels: 1, 2, and 3.
0
0
0
0
A.
B.
C.
D.
Fatou
Heron
Sierpinski
Sperner
Response
8a-sample-national-semifinals
Question #4—400 Points
0
0
0
0
60
The positive integers 725,725 and
432,432 have…
no common prime factors
B. exactly one common prime factor
C. exactly two common prime factors
D. exactly three common prime factors
A.
Response
8a-sample-national-semifinals
Question #5—500 Points
60
The San Antonio Spurs roster has 1 center, 7 guards,
4 forwards and 3 players who can play either center
or forward. How many different line-ups are possible,
consisting of 1 center, 2 guards, and 2 forwards?
0
A.
0
B.
0
C.
0
D.
504
1,386
1,764
2,205
Response
8a-sample-national-semifinals
Question #6—600 Points
60
According to its website, the San Antonio River Walk is five miles
long. Suppose that Rod runs at a constant rate of 8 miles per hour
and Sid runs at a constant rate of 6 miles per hour. Both Rod and
Sid start running at the same end of the River Walk and head
towards the other end. When Rod reaches the other end, he turns
around and runs back toward Sid. How long does it take for Rod
to meet Sid, measuring time from when they began the run?
0
0
0
0
A.
B.
C.
D.
2/3 hr.
3/4 hr.
5/7 hr.
11/14 hr.
Response
8a-sample-national-semifinals
Question #7—700 Points
90
What is the area of the region defined by
the inequality 𝑥𝑥 + 𝑦𝑦 + 𝑦𝑦 − 𝑥𝑥 ≤ 2 ?
0
A.
0
B.
0
C.
0
D.
2
3
2
2+ 2
Response
8a-sample-national-semifinals
𝑥𝑥 + 𝑦𝑦 + 𝑦𝑦 − 𝑥𝑥 ≤ 2
8a-sample-national-semifinals
Question #8—800 Points
120
What is the remainder when the 111-digit number
1234567891011…5960 is divided by 99?
0
A.
0
B.
0
C.
0
D.
0
30
48
51
Response
8a-sample-national-semifinals-end
Presented by
The American Mathematical Society
www.ams.org/wwtbam/
8b-sample-national-finals
𝜑𝜑𝜑s
8b-sample-national-finals
Question #1—1000 Points
A unit square is divided into four equal subsquares and the
upper left one is colored red. The lower right subsquare is
then divided into four equal smaller subsquares and the upper
left one is colored red. This dividing and coloring process
continues forever on the lower right subsquare constructed at
each step. How much area of the original square is colored red?
A.
B.
C.
D.
E.
1/4
5/16
1/3
5/12
1/2
60
8b-sample-national-finals
Question #2—1000 Points
How many ways are there to assign colors to the
vertices (nodes) of the graph below using white, green,
red, and blue, if two vertices that are connected by an
edge (line) must use different colors? Note: all colors
need not be used in a coloring.
A.
B.
C.
D.
E.
48
60
72
84
144
60
8b-sample-national-finals
Question #3—2000 Points
Consider a 27 × 21 rectangle subdivided into unit
squares (5 × 4 version shown below). How many unit
squares does a diagonal drawn from the lower left
corner to the upper right corner pass through?
(To pass through a unit square the diagonal must
contain interior points of the square.)
A.
B.
C.
D.
E.
27
36
42
45
48
60
8b-sample-national-finals
Question #4—2000 Points
Which of the following mathematicians
did not win a Fields Medal?
A. Timothy Gowers
B. John Nash
C. William Thurston
D. Edward Witten
E. Maryam Mirzakhani
60
8b-sample-national-finals
Question #5—3000 Points
Travis cashes his lawn-mowing check at a bank
where the absent-minded teller switches the
dollars and the cents. Travis then buys an ice cream
for 50¢ and has left twice the amount of the
original check. The amount of the original check
was:
A. less than $10
B. between $10 and $15
C. between $15 and $20
D. between $20 and $25
E. more than $25
60
8b-sample-national-finals
Question #6—3000 Points
Which of the following is largest?
A.
B.
C.
D.
E.
𝜋𝜋
𝜋𝜋
tan + cot
7
7
𝜋𝜋
𝜋𝜋
sin + cos
7
7
𝜋𝜋
𝜋𝜋
2tan + cos
7
7
𝜋𝜋
𝜋𝜋
2 sin cos
7
7
𝜋𝜋
𝜋𝜋
2 sin + cot
7
7
60
8b-sample-national-finals
Question #7—5000 Points
What is the sum of the squares of the
3
2
roots of 2𝑥𝑥 − 11𝑥𝑥 + 38𝑥𝑥 − 39 ?
A.
B.
C.
D.
E.
−45/2
−31/4
45/4
45/2
45
60
8b-sample-national-finals
2 + 19𝑥𝑥 − 39
2𝑥𝑥 3 − 11𝑥𝑥 2 + 38𝑥𝑥 − 39 → 𝑥𝑥 3 − 11
𝑥𝑥
2
2
If 𝑟𝑟, 𝑠𝑠, and 𝑡𝑡 are the roots, then
𝑟𝑟 + 𝑠𝑠 + 𝑡𝑡
2
=
𝑟𝑟 2
+ 𝑠𝑠 2
+ 𝑡𝑡 2
+ 2 𝑟𝑟𝑟𝑟 + 𝑟𝑟𝑟𝑟 + 𝑠𝑠𝑠𝑠 =
𝑟𝑟 2 + 𝑠𝑠 2 + 𝑡𝑡 2 + 2 19 = 121
4
31
𝑟𝑟 2 + 𝑠𝑠 2 + 𝑡𝑡 2 = 121
−
38
=
−
4
4
Polynomial Long Division and Root Power Sums,
Dan Kalman and Stacy Langton, Horizons,
February 2014
11 2
2
8b-sample-national-finals
Question #8—5000 Points
The earliest appearing four-digit number in
Pascal’s triangle is 1287, which is 𝐶𝐶 13, 5 .
What is the earliest appearing five-digit
number in Pascal’s triangle?
A.
B.
C.
D.
E.
10,000
10,720
11,380
11,440
12,870
60
8b-sample-national-finals-end
Presented by
The American Mathematical Society
www.ams.org/wwtbam/
9-sample-regional
Sample Regional Game
9-sample-regional
Question #1—100 Points
30
If we let P represent the population of
Rhode Island, then log10 𝑃𝑃 is about
0
0
0
0
A. −1
B. 0
C. 1
D. 6
Response
9-sample-regional
Question #2—200 Points
0
What is the surface area of a sphere of
radius 𝑟𝑟 divided by its volume?
A.
0
B.
0
C.
0
30
D.
1
3𝑟𝑟
4
3𝑟𝑟
3
𝑟𝑟
4
𝑟𝑟
Response
9-sample-regional
40
Question #3—300 Points
Three fair two-sided coins are flipped (and
H/T is noted). What is the probability that
not all three coins land with the same result?
0
A. 1/2
0
B.
0
0
5/8
C. 3/4
D. 7/8
Response
9-sample-regional
Question #4—400 Points
45
What is the measure of angle 𝛼𝛼 (in radians)?
A.
B.
C.
D.
tan−1
𝜋𝜋
4
1
2
𝛼𝛼
tan−1 2
It can’t be determined from the
given information.
Response
9-sample-regional
Question #5—500 Points
30
The set pictured is
named after which
of the following
mathematicians?
A.
B.
C.
D.
Rene Descartes
Pierre Fatou
Edward Lorenz
Benoît Mandelbrot
Response
9-sample-regional
The Mandelbrot Set
http://commons.wikimedia.org/wiki/File:Mandel
brot_set_rainbow_colors.png
9-sample-regional
Question #6—600 Points
60
The number 𝑛𝑛𝑛𝑛 is calculated for each positive
integer 𝑛𝑛 from 2010 to 2019 (inclusive). How
many different units/ones digits result?
0
A.
0
B.
0
C.
0
D.
6
7
8
9
Response
9-sample-regional
Question #7—700 Points
90
A ladder is leaning against the wall (as pictured). Then it is
adjusted (against the same wall) so that it now reaches twice
as high up the wall. The slope of the ladder is now …
A.
B.
C.
D.
Less than twice what it was initially
Exactly twice what it was initially
More than twice what it was initially
More information is needed to
determine which of the above is
correct.
Response
9-sample-regional
MAA Focus, “MAA Updates Testing for Calculus
Readiness,” Feb/Mar 2011,
Marilyn Carlson, Bernard Madison, and Rich West.
9-sample-regional
Question #8—800 Points
120
What is the sum of the digits of all solutions to
0
𝑥𝑥 + 𝑥𝑥 + 56 + 𝑥𝑥 − 𝑥𝑥 + 56 = 4 ?
A.
0
B.
0
C.
0
D.
8
11
17
25
Response
9-sample-regional
Off 2 Question
9-sample-regional
Square-Off Question
Analog clocks and watches are traditionally
displayed for sale showing the time 10:10. What is
the angle between the hour and minute hands at
that time (the smaller of the two angles)?
A.
B.
C.
D.
E.
11𝜋𝜋⁄
18
23𝜋𝜋⁄
36
2𝜋𝜋⁄
3
25𝜋𝜋⁄
36
3𝜋𝜋⁄
4
60
9-sample-regional
Two-Grand Prize
Question
9-sample-regional
180
Bonus Question
Given isosceles triangle ABC (below) where
AB = AC and M is the midpoint of AC. Find BM.
A.
B.
C.
D.
E.
𝐴𝐴
39�
2
57�
2
71�
2
89�
2
𝑀𝑀
5
3 7�
2
𝐵𝐵
4
𝐶𝐶
Response
9-sample-regional-answers
1.
2.
3.
4.
5.
D
C
C
B
D
Answers
6. C
7. C
8. A
Square-Off: B
Bonus: B
9-sample-regional-end
Presented by
The American Mathematical Society
www.ams.org/wwtbam/