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)

©2020 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
   • 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
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, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

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. On 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. On Saturdays only, there is a $10\mathrm{\%}$ discount on the price of a bag of avocados and a $20\mathrm{\%}$ 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. On 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. On 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 $y=\frac{1}{4}{x}^2$ 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.

   

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

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

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

   4. Determine 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 $n\ge 3$ vertices and $k\ge 0$ interior points with no three of these $n+k$ points lying on the same line,

   • no two line segments connecting pairs of these points cross anywhere except at their endpoints, and

   • each interior point is a vertex of at least one of the triangular regions.

   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, $T(6,1)=6$.

   

   

   1. What is the value of $T(3,2)$?

   2. Determine the value of $T(4,100)$.

   3. Determine the value of $n$ for which $T(n,n)=2020$.

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

   1. Show that $x_2-x_0$ is even for all possible values of $x_0$.

   2. Show that $x_{2026}-x_{2020}$ is divisible by 10 for all possible values of $x_0$.

   3. Parsa chooses an integer $n$ with $1\le n\le 100$ 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.

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• 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