2021 Hypatia Contest
(Grade 11)
April 2021
(in North America and South America)
April 2021
(outside of North American and South America)

©2021 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:
- 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, and 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 -intercepts of the graph of an equation like , you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.
Questions
A company rents out various sized passenger vehicles according to the following table. For example, a group of , or 8 people would need to rent a sports utility vehicle (SUV), which has a total cost of $200.00. Unfortunately, the total cost to rent a van is missing from the table. In each case, the members of the group equally share the total cost to rent the vehicle.
Vehicle |
Number of Passengers Required |
Total Cost |
Car |
1 to 4 |
$180.00 |
SUV |
5 to 8 |
$200.00 |
Van |
9 to 12 |
|
If 4 people rent a car, what is the cost per person?
If a group rents an SUV, what is the maximum possible cost per person?
When a van is rented, the difference between the maximum cost per person and the minimum cost per person is $6.00. Determine the total cost to rent a van.
Trapezoid has vertices , , , .
What is the area of trapezoid ?
The line passing through and intersects the -axis at the point . What are the coordinates of ?
The sides and are extended to intersect at the point . Determine the coordinates of .
Determine all possible points that lie on the line passing through and , so that the area of is 42.
The sequence , with terms , is defined by The sequence , with terms , is defined by For example, .
In this question, the following facts may be helpful:
A is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant called the common ratio. For example, is a geometric sequence with three terms and common ratio 2.
The sum of the first terms of a geometric sequence with first term , and common ratio , equals  .
What are the 5 terms for each sequence? That is, what are the values of and ?
For some real numbers and , for all . (You do not need to show this.) What are the values of and ?
Let be the sum of the first terms in sequence . That is, . Determine the smallest positive integer that satisfies .
In , the measure of is . Also, , , and hypotenuse has length . Further, the perimeter of is and the area of is .
If and , what are the values of and ?
If and , what is the value of ?
Determine all possible integer values of , and for which .
Suppose that , and are integers, that , and that for some prime number . Determine all possible values of .
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
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- Learn about our face-to-face workshops and our web resources
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- Investigate our online Master of Mathematics for Teachers
- Find your school's contest results