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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)

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©2018 University of Waterloo

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:

  1. SHORT ANSWER parts indicated by Lightbulb
  2. FULL SOLUTION parts indicated by Full Solution

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

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:
  1. Please read the instructions for the contest.
  2. Write all answers in the answer booklet provided.
  3. For questions marked Lightbulb, place your answer in the appropriate box in the answer booklet and show your work.
  4. For questions marked Full Solution, 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.

Questions

  1. Mr. Singh gives his students a test each week.

    1. LightbulbAneesh’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. LightbulbJon 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. Full solutionAfter 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.

    1. LightbulbOn Monday, Jessica drove at 90 km/h. How many minutes did it take Jessica to drive from Botown to Aville?

    2. LightbulbOn Tuesday, Jessica left Botown at 7:00 a.m.. What was the ETA displayed by her car at 7:00 a.m.?

    3. Full solutionOn Tuesday, Jessica drove at 90 km/h. Determine the ETA displayed by her car at 7:16 a.m..

    4. Full solutionOn 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. 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, \(T_{n+1}=1+T_1T_2T_3\cdots T_n\) for all integers \(n\geq 2\). For example, \(T_3=1+T_1T_2=3\).

    1. LightbulbWhat is the value of \(T_5\)?

    2. Full solutionProve that \(T_{n+1}=T_n^2-T_n+1\) for all integers \(n\geq 2\).

    3. Full solutionProve that \(T_n + T_{n+1}\) is a factor of \(T_nT_{n+1} - 1\) for all integers \(n\geq 2\).

    4. Full solutionProve that \(T_{2018}\) is not a perfect square.

    1. Full solutionConsider the two parabolas defined by the equations \(y=x^2-8x+17\) and \(y=-x^2+4x+7\).

      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 \(V_1PV_2Q\) is a parallelogram.

    2. Full solutionThe 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 \(S\).

      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 \(V_3RV_4S\) 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

For teachers...

Visit our website cemc.uwaterloo.ca to