2017 Hypatia Contest
(Grade 11)
Wednesday, April 12, 2017
(in North America and South America)
Thursday, April 13, 2017
(outside of North American and South America)

©2017 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 cyclic quadrilateral is a quadrilateral whose four vertices lie on some circle. In a cyclic quadrilateral, opposite angles add to . In the diagram, is a cyclic quadrilateral. Therefore, .

In Figure A below, is a cyclic quadrilateral.

If , what is the value of ?
In Figure B, and are cyclic quadrilaterals.

If , what is the value of and what is the value of ?
In Figure C, is a cyclic quadrilateral with and .

If , what is the value of ?
In Figure D, is a cyclic quadrilateral. is extended to , as shown.

If , determine the measure of in terms of .
A list of integers is written in a table, row after row from left to right. Row 1 has the integer 1. Row 2 has the integers and 3. Row has the consecutive integers beginning at 1 and ending at the odd integer. The first four rows are given below.
- Row 1: 1
- Row 2: 1, 2, 3
- Row 3: 1, 2, 3, 4, 5
- Row 4: 1, 2, 3, 4, 5, 6, 7
In the table, the integer to be written is 5, and it appears at the end of Row 3. In general, after having completed rows, a total of integers have been written.
What is the 25 integer written in the table and in which row does the 25 integer appear?
What is the 100 integer written in the table?
What is the 2017 integer written in the table?
In how many of the first 200 rows does the integer 96 appear?
The line intersects the parabola with equation at two points. What are the coordinates of these two points of intersection?
A line intersects the parabola with equation at and at . This line intersects the -axis at . Determine the value of .
A line intersects the parabola with equation at and at with . Determine the -intercept of this line.
For all , the curve intersects the parabola with equation at and at a second point whose coordinates depend on . All such points lie on a parabola. Determine the equation of this parabola.
A positive integer is called an -digit zigzag number if
,
the number’s digits are exactly (without repetition), and
for each group of three adjacent digits, either the middle digit is greater than each of the other two digits or the middle digit is less than each of the other two digits.
For example, 52314 is a 5-digit zigzag number but 52143 is not.
What is the largest 9-digit zigzag number?
Let be the number of -digit zigzag numbers with first digit and second digit greater than . Let be the number of -digit zigzag numbers with first digit and second digit less than .
Show that .
Show that equals
Determine the number of 8-digit zigzag numbers.
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
- 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