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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/