Hypatia Contest (Grade 11)

Transcription

1 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Hypatia Contest (Grade 11) Thursday, April 12, 2018 (in North America and South America) Friday, April 13, 2018 (outside of North America and South America) Time: 75 minutes 2018 University of Waterloo Do not open this booklet until instructed to do so. Number of questions: 4 Each question is worth 10 marks Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) previously stored information such as formulas, programs, notes, etc., (iv) a computer algebra system, (v) dynamic geometry software. Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express answers as simplified exact numbers except where otherwise indicated. For example, π + 1 and 1 2 are simplified exact numbers. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

2 NOTE: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 6. While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions and specific marks may be allocated for these steps. For example, while your calculator might be able to find the x-intercepts of the graph of an equation like y = x 3 x, you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down. 7. No student may write more than one of the Fryer, Galois and Hypatia Contests in the same year. 1. Mr. Singh gives his students a test each week. (a) Aneesh s scores on the first six tests were 17, 13, 20, 12, 18, and 10. What was the average (mean) of his test scores? (b) Jon scored 17 and 12 on his first two tests. After the third test, his average (mean) score was 14. What was his score on the third test? (c) After the first six tests, Dina had an average (mean) test score of 14. On each of the next n tests, Dina s score was 20 out of 20. After all of these tests, her average (mean) test score was 18. Determine the value of n. 2. Each day, Jessica drives from Botown to Aville, a distance of 120 km. During the drive, her car s navigation system constantly updates the estimated time of arrival (ETA) at Aville. The car predicts the ETA by assuming that Jessica will drive the remaining distance at 80 km/h. (a) On Monday, Jessica drove at 90 km/h. How many minutes did it take Jessica to drive from Botown to Aville? (b) On Tuesday, Jessica left Botown at 7:00 a.m.. What was the ETA displayed by her car at 7:00 a.m.? (c) On Tuesday, Jessica drove at 90 km/h. Determine the ETA displayed by her car at 7:16 a.m.. (d) On Wednesday, Jessica noted the ETA predicted by her car when she left Botown. She travelled the first part of the trip at 50 km/h and travelled the rest of the way at 100 km/h. Jessica arrived in Aville at the ETA predicted by her car when she left Botown. Determine the distance that she drove at a speed of 100 km/h.

3 3. A sequence T 1, T 2, T 3,... is defined by T 1 = 1, T 2 = 2, and each term after the second is equal to 1 more than the product of all previous terms in the sequence. That is, T n+1 = 1 + T 1 T 2 T 3 T n for all integers n 2. For example, T 3 = 1 + T 1 T 2 = 3. (a) What is the value of T 5? (b) Prove that T n+1 = Tn 2 T n + 1 for all integers n 2. (c) Prove that T n + T n+1 is a factor of T n T n+1 1 for all integers n 2. (d) Prove that T 2018 is not a perfect square. 4. (a) Consider the two parabolas defined by the equations y = x 2 8x + 17 and y = x 2 + 4x + 7. (i) Determine the coordinates of the vertices V 1 and V 2 of these two parabolas. (ii) Suppose that these two parabolas intersect at the points P and Q. Explain why the quadrilateral V 1 P V 2 Q is a parallelogram. (b) The two parabolas defined by the equations y = x 2 + bx + c and y = x 2 have vertices V 3 and V 4, respectively. For some values of b and c, these parabolas intersect at the points R and S. (i) Determine all pairs (b, c) for which the points R and S exist and the points V 3, V 4, R, S are distinct. (ii) Determine all pairs (b, c) for which the points R and S exist, the points V 3, V 4, R, S are distinct, and quadrilateral V 3 RV 4 S is a rectangle.

4 2018 Hypatia Contest (English) The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca For students... Thank you for writing the 2018 Hypatia Contest! Each year, more than students from more than 75 countries register to write the CEMC s Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website cemc.uwaterloo.ca to find Free copies of past contests Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests Information about careers in and applications of mathematics and computer science For teachers... Visit our website cemc.uwaterloo.ca to Obtain information about our 2018/2019 contests Register your students for the Canadian Senior and Intermediate Mathematics Contests which will be written in November Look at our free online courseware for senior high school students Learn about our face-to-face workshops and our web resources Subscribe to our free Problem of the Week Investigate our online Master of Mathematics for Teachers Find your school s contest results

5 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Hypatia Contest (Grade 11) Wednesday, April 12, 2017 (in North America and South America) Thursday, April 13, 2017 (outside of North America and South America) Time: 75 minutes 2017 University of Waterloo Do not open this booklet until instructed to do so. Number of questions: 4 Each question is worth 10 marks Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express answers as simplified exact numbers except where otherwise indicated. For example, π + 1 and 1 2 are simplified exact numbers. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

6 NOTE: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 6. While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions and specific marks may be allocated for these steps. For example, while your calculator might be able to find the x-intercepts of the graph of an equation like y = x 3 x, you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down. 7. No student may write more than one of the Fryer, Galois and Hypatia Contests in the same year. 1. A cyclic quadrilateral is a quadrilateral whose four vertices lie on some circle. In a cyclic quadrilateral, opposite angles add to 180. In the diagram, ABCD is a cyclic quadrilateral. Therefore, ABC + ADC = 180 = BAD + BCD. D A B (a) In Figure A below, ABCD is a cyclic quadrilateral. If BAD = 88, what is the value of u? (b) In Figure B, P QRS and ST QR are cyclic quadrilaterals. If ST Q = 58, what is the value of x and what is the value of y? (c) In Figure C, JKLM is a cyclic quadrilateral with JK = KL and JL = LM. If KJL = 35, what is the value of w? (d) In Figure D, DEF G is a cyclic quadrilateral. F G is extended to H, as shown. If DEF = z, determine the measure of DGH in terms of z. C D A 88º J P S D yº 35º K (2u)º C B T M 58º xº R wº zº E G L Q F Figure B Figure C Figure D H Figure A

7 2. A list of integers is written in a table, row after row from left to right. Row 1 has the integer 1. Row 2 has the integers 1, 2 and 3. Row n has the consecutive integers beginning at 1 and ending at the n th odd integer. In the table, the 9 th integer to be written is 5, and it appears at the end of Row 3. In general, after having completed n rows, a total of n 2 integers have been written. Row 1 1 Row Row Row (a) What is the 25 th integer written in the table and in which row does the 25 th integer appear? (b) What is the 100 th integer written in the table? (c) What is the 2017 th integer written in the table? (d) In how many of the first 200 rows does the integer 96 appear? 3. (a) The line y = 15 intersects the parabola with equation y = x 2 + 2x at two points. What are the coordinates of these two points of intersection? (b) A line intersects the parabola with equation y = x 2 3x at x = 4 and at x = a. This line intersects the y-axis at (0, 8). Determine the value of a. (c) A line intersects the parabola with equation y = x 2 + kx at x = p and at x = q with p q. Determine the y-intercept of this line. (d) For all k 0, the curve x = 1 k 3 y2 + 1 y intersects the parabola with equation k y = x 2 + kx at (0, 0) and at a second point T whose coordinates depend on k. All such points T lie on a parabola. Determine the equation of this parabola. 4. A positive integer is called an n-digit zigzag number if 3 n 9, the number s digits are exactly 1, 2,..., n (without repetition), and for each group of three adjacent digits, either the middle digit is greater than each of the other two digits or the middle digit is less than each of the other two digits. For example, is a 5-digit zigzag number but is not. (a) What is the largest 9-digit zigzag number? (b) Let G(n, k) be the number of n-digit zigzag numbers with first digit k and second digit greater than k. Let L(n, k) be the number of n-digit zigzag numbers with first digit k and second digit less than k. (i) Show that G(6, 3) = L(5, 3) + L(5, 4) + L(5, 5). (ii) Show that equals G(6, 1) + G(6, 2) + G(6, 3) + G(6, 4) + G(6, 5) + G(6, 6) L(6, 1) + L(6, 2) + L(6, 3) + L(6, 4) + L(6, 5) + L(6, 6). (c) Determine the number of 8-digit zigzag numbers.

8 2017 Hypatia Contest (English) The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca For students... Thank you for writing the 2017 Hypatia Contest! Each year, more than students from more than 60 countries register to write the CEMC s Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website cemc.uwaterloo.ca to find Free copies of past contests Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests Information about careers in and applications of mathematics and computer science For teachers... Visit our website cemc.uwaterloo.ca to Obtain information about our 2017/2018 contests Register your students for the Canadian Senior and Intermediate Mathematics Contests which will be written in November Look at our free online courseware for senior high school students Learn about our face-to-face workshops and our web resources Subscribe to our free Problem of the Week Investigate our online Master of Mathematics for Teachers Find your school s contest results

9 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Hypatia Contest (Grade 11) Wednesday, April 13, 2016 (in North America and South America) Thursday, April 14, 2016 (outside of North America and South America) Time: 75 minutes 2016 University of Waterloo Do not open this booklet until instructed to do so. Number of questions: 4 Each question is worth 10 marks Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as or , except where otherwise indicated. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

10 NOTE: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 6. While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions and specific marks may be allocated for these steps. For example, while your calculator might be able to find the x-intercepts of the graph of an equation like y = x 3 x, you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down. 7. No student may write more than one of the Fryer, Galois and Hypatia Contests in the same year. 1. Raisins are sold by the scoop, cup, jar, basket, or tub in the following proportions: 5 scoops of raisins fill 1 jar, 3 scoops of raisins fill 1 cup, 5 baskets of raisins fill 2 tubs, and 30 jars of raisins fill 1 tub. (a) How many tubs of raisins fill 30 baskets? (b) How many cups of raisins fill 6 jars? (c) Determine how many cups of raisins fill 1 basket. 2. If a line segment is drawn from the centre of a circle to the midpoint of a chord, it is perpendicular to that chord. For example, in Figure 1, OM is perpendicular to chord AB. If a line segment is drawn from the centre of a circle and is perpendicular to a chord, it passes through the midpoint of that chord. For example, in Figure 2, P R = QR. (a) In the diagram, a circle with radius 13 has a chord AB with length 10. If M is the midpoint of AB, what is the length of OM? Figure 1 Figure 2 A M O A B 13 M O P B O R Q (b) In a circle with radius 25, a chord is drawn so that its perpendicular distance from the centre of the circle is 7. What is the length of this chord? (c) In the diagram, the radius of the circle is 65. Two parallel chords ST and U V are drawn so that the perpendicular distance between the chords is 72 (MN = 72). If MN passes through the centre of the circle O, and ST has length 112, determine the length of UV. S U M O N T V

11 3. For a positive integer n, f(n) is defined as the exponent of the largest power of 3 that divides n. For example, f(126) = 2 since 126 = so 3 2 divides 126, but 3 3 does not. (a) What is the value of f(405)? (b) What is the value of f( )? (c) Let N be the positive integer equal to 100!. Determine the value of f(n). 50!20! (Note: If m is a positive integer, m! represents the product of the integers from 1 to m, inclusive. For example, 5! = = 120.) (d) Given that f(a) = 8 and f(b) = 7, determine all possible values of f(a + b). 4. Erin s Pizza (EP) and Lino s Pizza (LP) are located next door to each other. Each day, each of 100 customers buys one whole pizza from one of the restaurants. The price of a pizza at each restaurant is set each day and is always a multiple of 10 cents. If the two restaurants charge the same price, half of the 100 customers will go to each restaurant. For every 10 cents that one restaurant s price is higher than the other restaurant s price, it loses one customer to the other restaurant. The cost for each restaurant to make a pizza is $5.00. As an example, if EP charges $8.00 per pizza and LP charges $9.00 per pizza, the number of customers and the resulting profit for each restaurant is shown in the table below. Restaurant Price per pizza Number of customers Profit EP $ = ($8.00 $5.00) = $180 LP $ = ($9.00 $5.00) = $160 (a) On Monday, EP charges $7.70 for a pizza and LP charges $9.30. (i) (ii) How many customers does LP have? What is LP s total profit? (b) EP sets its price first and then LP sets its price. On Tuesday, EP charges $7.20 per pizza. What should LP s price be in order to maximize its profit? (c) On Wednesday, EP realizes what LP is doing: LP is maximizing its profit by setting its price after EP s price is set. EP continues to set its price first and sets a price that is a multiple of 20 cents. LP s price is still a multiple of 10 cents and the number of customers at each restaurant still follows the rule above. Determine the two prices that EP could charge in order to maximize its profit. State LP s profit in each case.

12 2016 Hypatia Contest (English) The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca For students... Thank you for writing the 2016 Hypatia Contest! Each year, more than students from more than 60 countries register to write the CEMC s Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website cemc.uwaterloo.ca to find Free copies of past contests Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests Information about careers in and applications of mathematics and computer science For teachers... Visit our website cemc.uwaterloo.ca to Obtain information about our 2016/2017 contests Register your students for the Canadian Senior and Intermediate Mathematics Contests which will be written in November Look at our free online courseware for senior high school students Learn about our face-to-face workshops and our web resources Subscribe to our free Problem of the Week Investigate our online Master of Mathematics for Teachers Find your school s contest results

13 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Hypatia Contest (Grade 11) Thursday, April 16, 2015 (in North America and South America) Friday, April 17, 2015 (outside of North America and South America) Time: 75 minutes 2015 University of Waterloo Do not open this booklet until instructed to do so. Number of questions: 4 Each question is worth 10 marks Calculators are allowed, with the following restriction: you may not use a device that has internet access, that can communicate with other devices, or that contains previously stored information. For example, you may not use a smartphone or a tablet. Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as or , except where otherwise indicated. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

14 NOTE: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 6. While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions and specific marks may be allocated for these steps. For example, while your calculator might be able to find the x-intercepts of the graph of an equation like y = x 3 x, you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down. 7. No student may write more than one of the Fryer, Galois and Hypatia Contests in the same year. 1. Each Hypatia Railway train has one engine car followed by some boxcars in a straight line. The distance between consecutive boxcars is 2 m. The distance between the engine car and the first boxcar is also 2 m. The engine car is 26 m in length and each boxcar is 15 m in length. The total length of a train is the distance from the front of the engine car to the end of the last boxcar. (a) What is the total length of a train with 10 boxcars? (b) A train has a total length of 2015 m. How many boxcars does the train have? (c) In the diagram, a southbound train with 14 boxcars crosses the border between Canada and the United States at a speed of 1.6 m/s. Determine the length of time in seconds during which a portion of the train is in Canada and a portion is in the United States at the same time. W N Canada United States train E 2. In the questions below, A, B, M, N, P, Q, and R are non-zero digits. (a) A two-digit positive integer AB equals 10A + B. For example, 37 = If AB BA = 72, what is the positive integer AB? (b) A two-digit positive integer M N is given. Explain why it is not possible that MN NM = 80. (c) A three-digit positive integer P QR equals 100P + 10Q + R. If P > R, determine the number of possible values of P QR RQP. S

15 3. Consider n line segments, where each pair of line segments intersect at a different point, and not at an endpoint of any of the n line segments. Let T (n) be the sum of the number of intersection points and the number of endpoints of the line segments. For example, T (1) = 2 and T (2) = 5. The diagram below illustrates that T (3) = 9. (a) What do T (4) and T (5) equal? (b) Express T (n) T (n 1) in terms of n. (c) Determine all possible values of n such that T (n) = Let gcd(a, b) represent the greatest common divisor of the two positive integers a and b. For example, gcd(18, 45) = 9 since 9 is the largest positive integer that divides both 18 and 45. The function P (n) is defined to equal the sum of the n greatest common divisors, gcd(1, n), gcd(2, n),..., gcd(n, n). For example: P (6) = gcd(1, 6) + gcd(2, 6) + gcd(3, 6) + gcd(4, 6) + gcd(5, 6) + gcd(6, 6) = = 15 Note: You may use the fact that P (ab) = P (a)p (b) for all positive integers a and b with gcd(a, b) = 1. (a) What is the value of P (125)? (b) If r and s are different prime numbers, prove that P (r 2 s) = r(3r 2)(2s 1). (c) If r and s are different prime numbers, prove that P (r 2 s) can never be equal to a power of a prime number (that is, can never equal t n for some prime number t and positive integer n). (d) Determine, with justification, two positive integers m for which P (m) = 243.

16 2015 Hypatia Contest (English) The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca For students... Thank you for writing the 2015 Hypatia Contest! Each year, more than students from more than 60 countries register to write the CEMC s Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website cemc.uwaterloo.ca to find Free copies of past contests Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests Information about careers in and applications of mathematics and computer science For teachers... Visit our website cemc.uwaterloo.ca to Obtain information about our 2015/2016 contests Register your students for the Canadian Senior and Intermediate Mathematics Contests which will be written in November Look at our free online courseware for senior high school students Learn about our face-to-face workshops and our web resources Subscribe to our free Problem of the Week Investigate our online Master of Mathematics for Teachers Find your school s contest results

17 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Hypatia Contest (Grade 11) Wednesday, April 16, 2014 (in North America and South America) Thursday, April 17, 2014 (outside of North America and South America) Do not open this booklet until instructed to do so University of Waterloo Time: 75 minutes Number of questions: 4 Calculators are permitted Each question is worth 10 marks Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as or , except where otherwise indicated. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published on our Web site, In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

18 TIPS: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 1. For real numbers a and b with a 0 and b 0, the operation is defined by a b = a + 4b. For example, 5 1 = 5 + 4(1) = 9 = 3. (a) What is the value of 8 7? (b) If 16 n = 10, what is the value of n? (c) Determine the value of (9 18) 10. (d) With justification, determine all possible values of k such that k k = k. 2. Each week, the MathTunes Music Store releases a list of the Top 200 songs. A new song Recursive Case is released in time to make it onto the Week 1 list. The song s position, P, on the list in a certain week, w, is given by the equation P = 3w 2 36w The week number w is always a positive integer. (a) What position does the song have on week 1? (b) Artists want their song to reach the best position possible. The closer that the position of a song is to position #1, the better the position. (i) What is the best position that the song Recursive Case reaches? (ii) On what week does this song reach its best position? (c) What is the last week that Recursive Case appears on the Top 200 list?

19 3. A pyramid ABCDE has a square base ABCD of side length 20. Vertex E lies on the line perpendicular to the base that passes through F, the centre of the base ABCD. It is given that EA = EB = EC = ED = 18. E (a) Determine the surface area of the pyramid ABCDE including its base. D 18 (b) Determine the height EF of the pyramid. (c) G and H are the midpoints of ED and EA, respectively. Determine the area of the quadrilateral BCGH. 4. The triple of positive integers (x, y, z) is called an Almost Pythagorean Triple (or APT) if x > 1 and y > 1 and x 2 + y 2 = z For example, (5, 5, 7) is an APT. (a) Determine the values of y and z so that (4, y, z) is an APT. (b) Prove that for any triangle whose side lengths form an APT, the area of the triangle is not an integer. (c) Determine two 5-tuples (b, c, p, q, r) of positive integers with p 100 for which (5t + p, bt + q, ct + r) is an APT for all positive integers t. C F B 20 A

20 2014 Hypatia Contest (English) The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca For students... Thank you for writing the 2014 Hypatia Contest! In 2013, more than students from around the world registered to write the Fryer, Galois and Hypatia Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website to find Free copies of past contests Workshops to help you prepare for future contests Information about our publications for mathematics enrichment and contest preparation For teachers... Visit our website to Obtain information about our 2014/2015 contests Learn about our face-to-face workshops and our resources Find your school contest results Subscribe to the Problem of the Week Read about our Master of Mathematics for Teachers program

21 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Hypatia Contest (Grade 11) Thursday, April 18, 2013 (in North America and South America) Friday, April 19, 2013 (outside of North America and South America) Do not open this booklet until instructed to do so University of Waterloo Time: 75 minutes Number of questions: 4 Calculators are permitted Each question is worth 10 marks Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as or , except where otherwise indicated. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published on our Web site, In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.

22 TIPS: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 1. At the JK Mall grand opening, some lucky shoppers are able to participate in a money giveaway. A large box has been filled with many $5, $10, $20, and $50 bills. The lucky shopper reaches into the box and is allowed to pull out one handful of bills. (a) Rad pulls out at least two bills of each type and his total sum of money is $175. What is the total number of bills that Rad pulled out? (b) Sandy pulls out exactly five bills and notices that she has at least one bill of each type. What are the possible sums of money that Sandy could have? (c) Lino pulls out six or fewer bills and his total sum of money is $160. There are exactly four possibilities for the number of each type of bill that Lino could have. Determine these four possibilities. 2. A parabola has equation y = (x 3) (a) What are the coordinates of the vertex of the parabola? (b) A new parabola is created by translating the original parabola 3 units to the left and 3 units up. What is the equation of the translated parabola? (c) Determine the coordinates of the point of intersection of these two parabolas. (d) The parabola with equation y = ax 2 + 4, a < 0, touches the parabola with equation y = (x 3) at exactly one point. Determine the value of a.

23 3. A sequence of m P s and n Q s with m > n is called non-predictive if there is some point in the sequence where the number of Q s counted from the left is greater than or equal to the number of P s counted from the left. For example, if m = 5 and n = 2 the sequence PPQQPPP is non-predictive because in counting the first four letters from the left, the number of Q s is equal to the number of P s. Also, the sequence QPPPQPP is non-predictive because in counting the first letter from the left, the number of Q s is greater than the number of P s. (a) If m = 7 and n = 2, determine the number of non-predictive sequences that begin with P. (b) Suppose that n = 2. Show that for every m > 2, the number of non-predictive sequences that begin with P is equal to the number of non-predictive sequences that begin with Q. (c) Determine the number of non-predictive sequences with m = 10 and n = (a) Twenty cubes, each with edge length 1 cm, are placed together in 4 rows of 5. What is the surface area of this rectangular prism? (b) A number of cubes, each with edge length 1 cm, are arranged to form a rectangular prism having height 1 cm and a surface area of 180 cm 2. Determine the number of cubes in the rectangular prism. (c) A number of cubes, each with edge length 1 cm, are arranged to form a rectangular prism having length l cm, width w cm, and thickness 1 cm. A frame is formed by removing a rectangular prism with thickness 1 cm located k cm from each of the sides of the original rectangular prism, as shown. Each of l, w and k is a positive integer. If the frame has surface area 532 cm 2, determine all possible values for l and w such that l w. k cm w cm k cm k cm k cm l cm 1 cm

24 2013 Hypatia Contest (English) The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca For students... Thank you for writing the 2013 Hypatia Contest! In 2012, more than students from around the world registered to write the Fryer, Galois and Hypatia Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website to find Free copies of past contests Workshops to help you prepare for future contests Information about our publications for mathematics enrichment and contest preparation For teachers... Visit our website to Obtain information about our 2013/2014 contests Learn about our face-to-face workshops and our resources Find your school contest results Subscribe to the Problem of the Week Read about our Master of Mathematics for Teachers program

25 The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Hypatia Contest (Grade 11) Thursday, April 12, 2012 (in North America and South America) Friday, April 13, 2012 (outside of North America and South America) Do not open this booklet until instructed to do so University of Waterloo Time: 75 minutes Number of questions: 4 Calculators are permitted Each question is worth 10 marks Parts of each question can be of two types: 1. SHORT ANSWER parts indicated by worth 2 or 3 marks each full marks given for a correct answer which is placed in the box part marks awarded only if relevant work is shown in the space provided 2. FULL SOLUTION parts indicated by worth the remainder of the 10 marks for the question must be written in the appropriate location in the answer booklet marks awarded for completeness, clarity, and style of presentation a correct solution poorly presented will not earn full marks WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED. Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages. Express calculations and answers as exact numbers such as π + 1 and 2, etc., rather than as or , except where otherwise indicated. Do not discuss the problems or solutions from this contest online for the next 48 hours. The name, grade, school and location of some top-scoring students will be published in the FGH Results on our Web site,

26 TIPS: 1. Please read the instructions on the front cover of this booklet. 2. Write all answers in the answer booklet provided. 3. For questions marked, place your answer in the appropriate box in the answer booklet and show your work. 4. For questions marked, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution. 5. Diagrams are not drawn to scale. They are intended as aids only. 1. Quadrilateral P QRS is constructed with QR = 51, as shown. The diagonals of P QRS intersect at 90 at point T, such that P T = 32 and QT = 24. Q P T 51 R S (a) Calculate the length of P Q. (b) Calculate the area of P QR. (c) If QS : P R = 12 : 11, determine the perimeter of quadrilateral P QRS. 2. (a) Determine the value of (a + b) 2, given that a 2 + b 2 = 24 and ab = 6. (b) If (x + y) 2 = 13 and x 2 + y 2 = 7, determine the value of xy. (c) If j + k = 6 and j 2 + k 2 = 52, determine the value of jk. (d) If m 2 + n 2 = 12 and m 4 + n 4 = 136, determine all possible values of mn.

27 y 3. (a) Points M( 1 2, 1 4 ) and N(n, n2 ) lie on the parabola with equation y = x 2, as shown. Determine the value of n such that MON = 90. y = x 2 N (n, n 2 ) 1 M (, 1 ) O (0, 0) 2 4 x (b) Points A(2, 4) and B(b, b 2 ) are the endpoints of a chord of the parabola with equation y = x 2, as shown. Determine the value of b so that ABO = 90. y y = x 2 A (2, 4) B (b, b 2 ) O (0, 0) x (c) Right-angled triangle P QR is inscribed in the parabola with equation y = x 2, as shown. Points P, Q and R have coordinates (p, p 2 ), (q, q 2 ) and (r, r 2 ), respectively. If p, q and r are integers, show that 2q + p + r = 0. y = x 2 y P R Q x 4. The positive divisors of 21 are 1, 3, 7 and 21. Let S(n) be the sum of the positive divisors of the positive integer n. For example, S(21) = = 32. (a) If p is an odd prime integer, find the value of p such that S(2p 2 ) = (b) The consecutive integers 14 and 15 have the property that S(14) = S(15). Determine all pairs of consecutive integers m and n such that m = 2p and n = 9q for prime integers p, q > 3, and S(m) = S(n). (c) Determine the number of pairs of distinct prime integers p and q, each less than 30, with the property that S(p 3 q) is not divisible by 24.

28 2012 Hypatia Contest (English) For students... The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Thank you for writing the 2012 Hypatia Contest! In 2011, more than students from around the world registered to write the Fryer, Galois and Hypatia Contests. Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November Visit our website to find Free copies of past contests Workshops to help you prepare for future contests Information about our publications for mathematics enrichment and contest preparation For teachers... Visit our website to Obtain information about our 2012/2013 contests Learn about our face-to-face workshops and our resources Find your school contest results Subscribe to the Problem of the Week Read about our Master of Mathematics for Teachers program

29 2011 Hypatia Contest (Grade 11) Wednesday, April 13, In the diagram, D and E are the midpoints of AB and BC respectively. y A (0, 6) D F B (0, 0) E C (8, 0) x (a) Determine an equation of the line passing through the points C and D. (b) Determine the coordinates of F, the point of intersection of AE and CD. (c) Determine the area of DBC. (d) Determine the area of quadrilateral DBEF. 2. A set S consists of all two-digit numbers such that: no number contains a digit of 0 or 9, and no number is a multiple of 11. (a) Determine how many numbers in S have a 3 as their tens digit. (b) Determine how many numbers in S have an 8 as their ones digit. (c) Determine how many numbers are in S. (d) Determine the sum of all the numbers in S. 3. Positive integers (x, y, z) form a Trenti-triple if 3x = 5y = 2z. (a) Determine the values of y and z in the Trenti-triple (50, y, z). (b) Show that for every Trenti-triple (x, y, z), y must be divisible by 6. (c) Show that for every Trenti-triple (x, y, z), the product xyz must be divisible by 900.

30 2011 Hypatia Contest Page 2 4. Let F (n) represent the number of ways that a positive integer n can be written as the sum of positive odd integers. For example, F (5) = 3 since 5 = = = 5 F (6) = 4 since 6 = = = = (a) Find F (8) and list all the ways that 8 can be written as the sum of positive odd integers. (b) Prove that F (n + 1) > F (n) for all integers n > 3. (c) Prove that F (2n) > 2F (n) for all integers n > 3.

31 2010 Hypatia Contest (Grade 11) Friday, April 9, Piravena must make a trip from A to B, then from B to C, then from C to A. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with C a distance of 3000 km from A and with B a distance of 3250 km from A. To take a bus, it costs Piravena $0.15 per kilometre. To take an airplane, it costs her a $100 booking fee, plus $0.10 per kilometre. B C 3250 km 3000 km A (a) To begin her trip she flew from A to B. Determine the cost to fly from A to B. (b) Determine the distance she travels for her complete trip. (c) Piravena chose the least expensive way to travel between cities and her total cost was $ Given that she flew from A to B, determine her method of transportation from B to C and her method of transportation from C to A. 2. A function f is such that f(x) f(x 1) = 4x 9 and f(5) = 18. (a) Determine the value of f(6). (b) Determine the value of f(3). (c) If f(x) = 2x 2 + px + q, determine the values of p and q. 3. In the diagram, square ABCD has sides of length 4, and ABE is equilateral. Line segments BE and AC intersect at P. Point Q is on BC so that P Q is perpendicular to BC and P Q = x. (a) Determine the measures of the angles of BP C. A B (b) Find an expression for the length of BQ in terms of x. P x Q (c) Determine the exact value of x. (d) Determine the exact area of AP E. D E C

32 2010 Hypatia Contest Page 2 4. (a) Determine all real values of x satisfying the equation x 4 6x = 0. (b) Determine the smallest positive integer N for which x x 2 + N can be factored as (x 2 + rx + s)(x 2 + tx + u) with r, s, t, u integers and r 0. (c) Prove that x 4 + Mx 2 + N cannot be factored as in (b) for any integers M and N with N M = 37.

33 2009 Hypatia Contest (Grade 11) Wednesday, April 8, Emma counts the number of students in her class with each eye and hair colour, and summarizes the results in the following table: Hair Colour Brown Blonde Red Blue Eye Colour Green Brown (a) What percentage of the students have both green eyes and brown hair? (b) What percentage of the students have green eyes or brown hair or both? (c) Of the students who have green eyes, what percentage also have red hair? (d) Determine how many students with red hair must join the class so that the percentage of the students in the class with red hair becomes 36%. 2. An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant d, called the common difference. For example, the sequence 2, 11, 20, 29, 38 is an arithmetic sequence with five terms and a common difference of d = 9. (a) An arithmetic sequence has three terms. The three terms add to 180. Determine the middle term of this sequence. (b) An arithmetic sequence has five terms. The five terms add to 180. Show that at least one of the five terms equals 36. (c) An arithmetic sequence has six terms. The six terms in the sequence add to 180. Determine the sum of the first and sixth terms of the sequence. 3. Triangle ABC has vertices A(0, 8), B(2, 0), C(8, 0). (a) Determine the equation of the line through B that cuts the area of ABC in half. (b) A vertical line intersects AC at R and BC at S, forming RSC. If the area of RSC is 12.5, determine the coordinates of point R. (c) A horizontal line intersects AB at T and AC at U, forming AT U. If the area of AT U is 13.5, determine the equation of the horizontal line.