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2020 Hypatia Contest
(Grade 11)

Wednesday, April 15, 2020
(in North America and South America)

Thursday, April 16, 2020
(outside of North American and South America)

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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. At a local grocery store, avocados are sold for $5.00 per bag and mangoes for $12.50 per box. A bag contains \(6\) avocados and a box contains \(15\) mangoes. Only a whole number of bags and a whole number of boxes can be purchased.

    1. LightbulbOn Friday, a chef purchased \(12\) bags of avocados and some boxes of mangoes. If the total cost was $135.00, how many boxes of mangoes were purchased?

    2. LightbulbOn Saturdays only, there is a \(10\%\) discount on the price of a bag of avocados and a \(20\%\) discount on the price of a box of mangoes. What is the total cost for \(8\) bags of avocados and \(4\) boxes of mangoes on Saturdays?

    3. Full solutionOn Monday, the chef needed \(100\) avocados and \(70\) mangoes. The chef purchased just enough bags and boxes. Determine how much the purchase cost her.

    4. Full solutionOn Tuesday, the chef made special tarts that each required \(1\) avocado and \(2\) mangoes. If the chef spent exactly $75.00 on avocados and mangoes, determine the greatest number of tarts that she could have made.

  2. The parabola with equation \(y=\frac 14x^2\) has its vertex at the origin and the \(y\)-axis as its axis of symmetry. For any point \((p, q)\) on the parabola (not at the origin), we can form a parabolic rectangle. This rectangle will have one vertex at \((p, q)\), a second vertex on the parabola, and the other two vertices on the \(x\)-axis. A parabolic rectangle with area \(4\) is shown.

    A shaded rectangle is drawn on the Cartesian Plane so that the bottom edge lies along the x-axis. The rectangle covers the lower part of the parabola. The upper-right of the rectangle intersects the parabola and the point (2,1), and the upper-left of the rectangle intersects the parabola at an unlabelled point in the second quadrant.

    1. LightbulbA parabolic rectangle has one vertex at \((6, 9)\). What are the coordinates of the other three vertices?

    2. LightbulbWhat is the area of the parabolic rectangle having one vertex at \((-3, 0)\)?

    3. Full solutionDetermine the areas of the two parabolic rectangles that have a side length of 36.

    4. Full solutionDetermine the area of the parabolic rectangle whose length and width are equal.

  3. A triangulation of a regular polygon is a division of its interior into triangular regions. In such a division, each vertex of each triangle is either a vertex of the polygon or an interior point of the polygon. In a triangulation of a regular polygon with \(n\geq 3\) vertices and \(k\geq 0\) interior points with no three of these \(n+k\) points lying on the same line,

    Every regular polygon has at least one triangulation. The number of triangles formed by any triangulation of a regular polygon with \(n\) vertices and \(k\) interior points is constant and is denoted \(T(n, k)\). For example, in every possible triangulation of a regular hexagon and one interior point, there are exactly 6 triangles. That is, \(T(6,1)=6\).

    Illustration of \(T(6,0)=4\)
    Th e interior of a regular hexagon is divided into four triangular regions by connecting three pairs of vertices.
    Illustration of \(T(6,1)=6\)
    A regular hexagon has one interior point. This interior point is connected to each vertex of the hexagon and divides the interior of the hexagon int0 six triangular regions. A second regular hexagon also has one interior point. If the vertices are numbered in order, then this interior point is connected to vertices 1, 2, 3, and 5 and vertex 5 is connected to vertices 1 and 3, dividing the interior of the hexagon into six triangular regions.

    1. LightbulbWhat is the value of \(T(3, 2)\)?

    2. Full solutionDetermine the value of \(T(4, 100)\).

    3. Full solutionDetermine the value of \(n\) for which \(T(n,n) = 2020\).

  4. Let \(x_0\) be a non-negative integer. For each integer \(i\geq 0\), define \(x_{i+1}=(x_i)^2+1\).

    1. Full solutionShow that \(x_2-x_0\) is even for all possible values of \(x_0\).

    2. Full solutionShow that \(x_{2026}-x_{2020}\) is divisible by 10 for all possible values of \(x_0\).

    3. Full solutionParsa chooses an integer \(n\) with \(1\leq n\leq 100\) at random and sets \(x_0=n\). Determine the probability that \(x_{115}-110\) is divisible by 105.

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