2018 Hypatia Contest
(Grade 11)

Thursday, April 12, 2018
(in North America and South America)

Friday, April 13, 2018
(outside of North American and South America)

University of Waterloo Logo

Instructions

Time: 75 minutes

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) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.

Parts of each question can be of two types:

SHORT ANSWER parts indicated by
- worth 2 or 3 marks each
- full marks are given for a correct answer which is placed in the box
- part marks are awarded if relevant work is shown in the space provided
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 $1 - \sqrt{2}$ 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, and score range 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.

NOTE:

Please read the instructions for the contest.
Write all answers in the answer booklet provided.
For questions marked , place your answer in the appropriate box in the answer booklet and show your work.
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.
Diagrams are not drawn to scale. They are intended as aids only.
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.

Questions

Mr. Singh gives his students a test each week.
1. 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?
2. 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?
3. 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 $n$ .
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.
1. On Monday, Jessica drove at 90 km/h. How many minutes did it take Jessica to drive from Botown to Aville?
2. On Tuesday, Jessica left Botown at 7:00 a.m.. What was the ETA displayed by her car at 7:00 a.m.?
3. On Tuesday, Jessica drove at 90 km/h. Determine the ETA displayed by her car at 7:16 a.m..
4. 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.
A sequence $T_{1}, T_{2}, T_{3}, \dots$ 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, $T_{n + 1} = 1 + T_{1} T_{2} T_{3} \dots T_{n}$ for all integers $n \geq 2$ . For example, $T_{3} = 1 + T_{1} T_{2} = 3$ .
1. What is the value of $T_{5}$ ?
2. Prove that $T_{n + 1} = T_{n}^{2} - T_{n} + 1$ for all integers $n \geq 2$ .
3. Prove that $T_{n} + T_{n + 1}$ is a factor of $T_{n} T_{n + 1} - 1$ for all integers $n \geq 2$ .
4. Prove that $T_{2018}$ is not a perfect square.
1. Consider the two parabolas defined by the equations $y = x^{2} - 8 x + 17$ and $y = - x^{2} + 4 x + 7$ .
  1. Determine the coordinates of the vertices $V_{1}$ and $V_{2}$ of these two parabolas.
  2. 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.
2. The two parabolas defined by the equations $y = - x^{2} + b x + c$ 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 $S$ .
  1. Determine all pairs $(b, c)$ for which the points $R$ and $S$ exist and the points $V_{3}$ , $V_{4}$ , $R$ , $S$ are distinct.
  2. 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 $V_{3} R V_{4} S$ is a rectangle.

Further Information

For students...

Thank you for writing the Hypatia Contest!

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 future contests
Look at our free online courseware for 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