The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
Hypatia Contest
(Grade 11)
Wednesday, April 5, 2023
(in North America and South America)
Thursday, April 6, 2023
(outside of North America and South America)
Time:
75 minutes
©2023 University of Waterloo
Do not open this booklet until instructed to do so.
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 given for a correct answer which is placed in the box
• part marks awarded only 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 must be supplied by your supervising teacher
and 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.
p
For example, ⇡ + 1 and 1
2 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 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 on the front cover of this booklet.
2. Write all answers in the answer booklet provided.
3.
For questions marked
, place your answer in the appropriate box in the answer
booklet and show your work.
4.
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 x-intercepts
of the graph of an equation like y = x3 x, you should show the algebraic steps that
you used to find these numbers, rather than simply writing these numbers down.
No student may write more than one of the Fryer, Galois and Hypatia Contests in the
same year.
5.
6.
7.
1.
A game is played in which each throw of a ball lands in one of two holes: the closer hole
or the farther hole. A throw landing in the closer hole scores 2 points, while a throw
landing in the farther hole scores 5 points. A player’s total score is equal to the sum of
the scores on their throws.
(a) Jasmin had 3 throws that each scored 2 points and 4 throws that each scored
5 points. What was Jasmin’s total score?
(b) Sam had twice as many throws that scored 2 points as throws that scored 5 points.
If Sam’s total score was 36 points, how many throws did Sam take?
(c) Tia had t throws that each scored 2 points and f throws that each scored 5 points.
If Tia’s total score was 37 points, determine all possible ordered pairs (t, f ).
(d) The game is changed so that each throw scores 6 or 21 points instead of 2 or 5.
Explain whether or not it is possible to have a total score of 182 points.
2.
In each question below, ABCD is a rectangle with AB = 2 and AD = 15.
(a) Point E is on BC, as shown. What is the total area of the shaded regions?
E
B
C
2
A
15
D
(b) Point F is on BC, and BD intersects AF at G, as shown. If the area of 4F GD
is 5, what is the area of the shaded region?
C
F
B
2
G
A
D
15
(c) Point P is on BC and R is on AD. BR and AP intersect at S and P D and RC
intersect at Q, as shown. If the area of P QRS is 6, determine the total area of
the shaded regions.
2
A
C
P
B
Q
S
R
D
3.
For any positive integer with three or more di↵erent, non-zero digits, let a cousin be
defined as the result of switching two digits of the integer. For example, the integer 156
has three cousins:
• 516 (obtained by switching the 1st and 2nd digits),
• 651 (obtained by switching the 1st and 3rd digits), and
• 165 (obtained by switching the 2nd and 3rd digits).
(a) In no particular order, five of the six cousins of 6238 are listed below. Which
cousin is missing from the list?
2638
3268
8236
6328
6283
(b) In no particular order, the following list contains an original integer as well as all
of its cousins. What is the original integer?
726 194
276 491
729 461
126 497
726 941
926 471
716 492
721 496
746 291
796 421
762 491
426 791
627 491
726 419
726 491
724 691
(c) Suppose that c and d are distinct, non-zero digits. The three-digit integer cd3
minus one of its cousins is equal to the three-digit integer d95. Determine the
values of c and d and show that no other values are possible.
(d) Suppose that m and n are distinct, non-zero digits. The sum of the six cousins of
the four-digit integer mn97 is equal to the five-digit integer nmnm7. Determine
the values of m and n and show that no other values are possible.
4.
The Great Math Company has a random integer generator which produces an integer
from 1 to 9 inclusive, where each integer is generated with equal probability. Each
member of the Multiplication Team uses this generator a certain number of times and
then calculates the product of their integers.
(a) Amarpreet uses the generator 3 times. What is the probability that the product
is a prime number?
(b) Braxton uses the generator 4 times. Determine the probability that the product
is divisible by 5, but not divisible by 7.
(c) Camille uses the generator 2023 times. Let p be the probability that the product
is not divisible by 6. Determine the ones digit of the integer equal to p ⇥ 92023 .
2023
Hypatia
Contest
(English)
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
For students...
Thank you for writing the 2023 Hypatia Contest! Each year, more
than 260 000 students from more than 80 countries register to write
the CEMC’s Contests.
Encourage your teacher to register you for the Canadian Intermediate
Mathematics Contest or the Canadian Senior Mathematics Contest,
which will be written in November 2023.
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 our 2023/2024 contests
• Register your students for the Canadian Senior and Intermediate
Mathematics Contests which will be written in November
• Look at our free online courseware for senior 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
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
Hypatia Contest
(Grade 11)
Tuesday, April 12, 2022
(in North America and South America)
Wednesday, April 13, 2022
(outside of North America and South America)
Time:
75 minutes
©2022 University of Waterloo
Do not open this booklet until instructed to do so.
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 given for a correct answer which is placed in the box
• part marks awarded only 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 must be supplied by your supervising teacher
and 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.
p
For example, ⇡ + 1 and 1
2 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 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 on the front cover of this booklet.
2. Write all answers in the answer booklet provided.
3.
For questions marked
, place your answer in the appropriate box in the answer
booklet and show your work.
4.
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 x-intercepts
of the graph of an equation like y = x3 x, you should show the algebraic steps that
you used to find these numbers, rather than simply writing these numbers down.
No student may write more than one of the Fryer, Galois and Hypatia Contests in the
same year.
5.
6.
7.
Useful Fact:
It may be helpful to know that 2n
1.
n + 1 for all positive integers n.
A regular hexagon is a polygon that has six sides with equal length and six interior
angles with equal measure. In Figure 1, regular hexagon ABCDEF has side length 2x
and its vertices lie on the circle with centre O. The diagonals AD, BE and CF divide
ABCDEF into six congruent equilateral triangles.
(a) In terms of x, what is the radius of the circle?
(b) The midpoint of side AB is labelled M , as shown in Figure 2. In terms of x,
what is the length of OM ?
(c) In terms of x, what is the area of hexagon ABCDEF ?
(d) The region that lies inside the circle and outside hexagon ABCDEF is shaded,
as shown in Figure 3. The area of this shaded region is 123. Rounded to the
nearest tenth, determine the value of x.
A
F
A
F
A
F
M
O
B
C
E
D
Figure 1
O
B
C
E
D
Figure 2
O
B
C
E
D
Figure 3
2.
With 1 kg of muffin batter, exactly 24 mini muffins and 2 large muffins can be made.
With 2 kg of muffin batter, exactly 36 mini muffins and 6 large muffins can be made.
(a) With 2 kg of muffin batter, exactly 48 mini muffins and n large muffins can also
be made. What is the value of n?
(b) With x kg of muffin batter, exactly 84 mini muffins and 10 large muffins can be
made. What is the value of x?
(c) Determine how many mini muffins can be made using the same amount of batter
that is needed to make 7 large muffins.
3.
A sequence is created in such a way that
• a real number is chosen as the first number in the sequence, and
• each of the following numbers in the sequence is generated by applying a function
to the previous number in the sequence.
For example, if the first number in a sequence is 1 and the following numbers are
generated by the function x2 5, then the first three numbers in the sequence are 1, 4
and 11 since 12 5 = 4 and ( 4)2 5 = 11.
(a) The first number in a sequence is 3 and the sequence is generated by the function
x2 3x + 1. What are the first four numbers in the sequence?
(b) The number 7 is the third number in a sequence generated by the function
x2 4x + 7. What are all possible first numbers in the sequence?
(c) The first number in a sequence is c and the sequence is generated by the function
x2 7x 48. If all numbers in the sequence are equal to c, determine all possible
values of c.
(d) A sequence generated by the function x2 12x + 39 alternates between two
di↵erent numbers. That is, the sequence is a, b, a, b, a, b, . . . , with a 6= b.
Determine all possible values of a.
4.
Every integer N > 1 can be written as N = pa11 pa22 pa33 · · · pakk , where k is a positive
integer, p1 < p2 < p3 < · · · < pk are prime numbers, and a1 , a2 , a3 , . . . , ak are
positive integers. For example, 1400 = 23 52 71 .
The number of positive divisors of N is denoted by f (N ). It is known that
f (N ) = (1 + a1 )(1 + a2 )(1 + a3 ) · · · (1 + ak )
(a) How many positive divisors does 240 have? That is, what is the value of f (240)?
(b) Define an integer N > 1 to be refactorable if it is divisible by f (N ). For example,
both 6 and 8 have 4 positive divisors, so 8 is refactorable and 6 is not refactorable.
This is because 8 is divisible by 4, but 6 is not divisible by 4. Determine all
refactorable numbers N with f (N ) = 6.
(c) Determine the smallest refactorable number N with f (N ) = 256.
(d) Show that for every integer m > 1, there exists a refactorable number N such
that f (N ) = m.
2022
Hypatia
Contest
(English)
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
For students...
Thank you for writing the 2022 Hypatia Contest! Each year, more
than 260 000 students from more than 80 countries register to write
the CEMC’s Contests.
Encourage your teacher to register you for the Canadian Intermediate
Mathematics Contest or the Canadian Senior Mathematics Contest,
which will be written in November 2022.
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 our 2022/2023 contests
• Register your students for the Canadian Senior and Intermediate
Mathematics Contests which will be written in November
• Look at our free online courseware for senior 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
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
Hypatia Contest
(Grade 11)
April 2021
(in North America and South America)
April 2021
(outside of North America and South America)
Time:
75 minutes
©2021 University of Waterloo
Do not open this booklet until instructed to do so.
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 given for a correct answer which is placed in the box
• part marks awarded only 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 must be supplied by your supervising teacher
and 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.
p
For example, ⇡ + 1 and 1
2 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 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 on the front cover of this booklet.
2. Write all answers in the answer booklet provided.
3.
For questions marked
, place your answer in the appropriate box in the answer
booklet and show your work.
4.
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 x-intercepts
of the graph of an equation like y = x3 x, you should show the algebraic steps that
you used to find these numbers, rather than simply writing these numbers down.
No student may write more than one of the Fryer, Galois and Hypatia Contests in the
same year.
5.
6.
7.
1.
A company rents out various sized passenger vehicles according to the following table. For
example, a group of 5, 6, 7, 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
Car
SUV
Van
Number of Passengers Required
1 to 4
5 to 8
9 to 12
Total Cost
$180.00
$200.00
(a) If 4 people rent a car, what is the cost per person?
(b) If a group rents an SUV, what is the maximum possible cost per person?
(c) When a van is rented, the di↵erence between the maximum cost per person and
the minimum cost per person is $6.00. Determine the total cost to rent a van.
2.
Trapezoid ABCD has vertices A(0, 0), B(12, 0), C(11, 5), D(2, 5).
(a) What is the area of trapezoid ABCD?
(b) The line passing through B and D intersects the y-axis at the point E. What
are the coordinates of E?
(c) The sides AD and BC are extended to intersect at the point F . Determine the
coordinates of F .
(d) Determine all possible points P that lie on the line passing through B and D, so
that the area of 4P AB is 42.
3.
The sequence A, with terms a1 , a2 , a3 , . . ., is defined by
an = 2n , for n 1.
The sequence B, with terms b1 , b2 , b3 , . . ., is defined by
b1 = 1, b2 = 1, and bn = bn 1 + 2bn 2 , for n
3.
For example, b3 = b2 + 2b1 = 1 + 2(1) = 3.
In this question, the following facts may be helpful:
• A geometric sequence 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, 3, 6, 12 is a geometric sequence with three terms and common
ratio 2.
• The sum of the first n terms of a geometric sequence with first term a, and common
◆
✓
1 rn
.
ratio r 6= 1, equals a
1 r
(a) What are the 5th terms for each sequence? That is, what are the values of a5
and b5 ?
(b) For some real numbers p and q, bn = p · (an ) + q · ( 1)n for all n
not need to show this.) What are the values of p and q?
4.
1. (You do
(c) Let Sn be the sum of the first n terms in sequence B.
That is,
Sn = b1 + b2 + b3 + · · · + bn . Determine the smallest positive integer n that
satisfies Sn 162021 .
In 4XY Z, the measure of \XZY is 90 . Also, Y Z = x cm, XZ = y cm, and hypotenuse
XY has length z cm. Further, the perimeter of 4XY Z is P cm and the area of 4XY Z
is A cm2 .
(a) If x = 20 and y = 21, what are the values of A and P ?
(b) If z = 50 and A = 336, what is the value of P ?
(c) Determine all possible integer values of x, y and z for which A = 3P .
(d) Suppose that x, y and z are integers, that P = 510, and that A = kP for some
prime number k. Determine all possible values of k.
2021
Hypatia
Contest
(English)
The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
For students...
Thank you for writing the 2021 Hypatia Contest! Each year, more
than 260 000 students from more than 80 countries register to write
the CEMC’s Contests.
Encourage your teacher to register you for the Canadian Intermediate
Mathematics Contest or the Canadian Senior Mathematics Contest,
which will be written in November 2021.
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 our 2021/2022 contests
• Register your students for the Canadian Senior and Intermediate
Mathematics Contests which will be written in November
• 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