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2019 Hypatia Contest
(Grade 11)

Wednesday, April 10, 2019
(in North America and South America)

Thursday, April 11, 2019
(outside of North American and South America)

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©2019 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. A rectangular piece of metal measures 91 cm by 16 cm. Four identical circular discs are punched out of this piece of metal. The centres of the circular holes are on the midline of the rectangle, \(AJ\), as shown. These four holes are equally spaced along the piece of metal. That is, \(AB=CD\), for example.

    A and J are the midpoints of the 16cm sides. B and C mark the diameter of the first circle along the midline AJ. D marks the start of the second circle at the midline AJ.

    1. LightbulbIf the radius of each hole is 2 cm, what is the distance along the midline between adjacent holes (i.e. what is the length of \(CD\))?

    2. LightbulbIf the distance along the midline between adjacent holes is equal to the radius of each hole, what is the radius of each hole?

    3. Full solutionShow why the fact that holes must be circles means that the distance between adjacent holes cannot be 5 cm.

  2. A bump can be added to any line segment through the following process:

    The series of diagrams below shows a bump being added to a line segment of length 3, transforming it into a path of length 4.

    A transformation is represented by a series of four diagrams, with an arrow pointing from one diagram to the next. The first diagram is a horizontal line segment of length 3. In the second diagram the line segment has been divided into three smaller line segments of length 1. In the third diagram, the middle line segment of length 1, is replaced with a dashed line. In the fourth diagram, the dashed line is removed and two new line segments of length 1 have been added, creating a triangular bump above the gap, and joining the two endpoints of the horizontal line segments. The four line segments are all connected.

    1. LightbulbA line segment has length 21. How long will the path be after a bump is added?

    2. LightbulbA path with exactly one bump has length 240. How long was the original line segment?

    3. Full solutionLin starts with a line segment that has length 36 and adds a bump to it. She then adds bumps to each line segment of that path. The resulting figure is shown below on the right.

      Line segment AD has length 36. A bump is added to the line segment AD resulting in a path of four segments of equal length. A bump is then added to each of these four segments, resulting in a path of sixteen line segments of equal length.

      What is the total path length of the resulting figure?

    4. Full solutionAnn starts with a line segment having length equal to some positive integer \(n\) and adds a bump to it resulting in Path 1. Ann then adds bumps to each line segment of Path 1 resulting in Path 2. She continues this process to create Path 3, Path 4, and finally Path 5. If the length of Path 5 is an integer, determine the smallest possible value of \(n\).

  3. The arithmetic mean of two positive real numbers \(x\) and \(y\) is half the sum of the two numbers, or \(\dfrac{x+y}{2}\). The geometric mean of two positive real numbers \(x\) and \(y\) is the square root of the product of the two numbers, or \(\sqrt{xy}\).

    1. LightbulbWhat are the arithmetic and geometric means of 36 and 64?

    2. LightbulbDetermine a pair of positive real numbers whose arithmetic mean is 13 and geometric mean is 12.

    3. Full solutionFor two positive integers \(x\) and \(y\), the arithmetic mean minus the geometric mean is equal to 1. Determine, with justification, all such pairs \((x, y)\) where \(x < y \leq 50\).

    1. LightbulbSuppose that \(c\) is a real number. Solve the following system of equations for \(x\) and \(y\) in terms of \(c\): \[\begin{aligned} 3x + 4y & = 10 \\ 5x + 6y & = c\end{aligned}\]

    2. Full solutionDetermine all integers \(d\) for which the system of equations \[\begin{aligned} x + 2y & = 3 \\ 4x + dy & = 6\end{aligned}\] has a solution \((x,y)\), where \(x\) and \(y\) are integers.

    3. Full solutionDetermine a positive integer \(k\) for which there are exactly 8 integers \(n\) for which the system of equations \[\begin{aligned} (9n+6)x - (3n+2)y & = 3n^2+6n+(3k+5) \\ (6n+4)x + (3n^2+2n)y & = - n^2 + (2k+2)\end{aligned}\] has a solution \((x,y)\), where \(x\) and \(y\) are integers.

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