2024 Pascal Contest
(Grade 9)
Wednesday, February 28, 2024
(in North America and South America)
Thursday, February 29, 2024
(outside of North American and South America)
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©2023 University of Waterloo
Instructions
Time: 60 minutes
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.
- Do not open the Contest booklet until you are told to do so.
- You may use rulers, compasses and paper for rough work.
- Be sure that you understand the coding system for your response form. If you are not sure, ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in circles completely.
- On your response form, print your school name and city/town in the box in the upper right corner.
- Be certain that you code your name, age, grade, and the Contest you are writing in the response form. Only those who do so can be counted as eligible students.
- Part A and Part B of this contest are multiple choice. Each of the questions in these parts
is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. After making your choice, fill in the appropriate circle on the response form.
- The correct answer to each question in Part C is an integer from 0 to 99, inclusive. After
deciding on your answer, fill in the appropriate two circles on the response form. A one-digit
answer (such as "7") must be coded with a leading zero ("07").
- Scoring:
- Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
- There is no penalty for an incorrect answer.
- Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
- Diagrams are not drawn to scale. They are intended as aids only.
- When your supervisor tells you to begin, you will have sixty minutes of working time.
- You may not write more than one of the Pascal, Cayley and Fermat Contests in any given year.
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 the website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some students may be shared with other mathematical organizations for other recognition opportunities.
Scoring:
- There is no penalty for an incorrect answer.
- Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
Part A: Each correct answer is worth 5.
The value of
is
The integers and are shown on a number line.
The distance between them is
In the diagram, the word PASCAL is written inside a circle.
When the diagram is rotated in the clockwise direction,
the resulting figure is





In a certain year, July 1 was a Wednesday. In that year, July 17
was a
Six rhombi of side length
are arranged as shown.
What is the perimeter of this figure?
Narsa buys a package of
cookies on Monday morning. The bar graph shows the number of cookies
that Narsa eats each day from Monday to Friday.
How many cookies are left in the package after Friday?
Shuxin begins with red
candies, yellow candies, and
blue candies. After eating some
of the candies, there are equal numbers of red, yellow, and blue candies
remaining. What is the smallest possible number of candies that Shuxin
ate?
There are students in a
class. In total, of them have
black hair, of them wear glasses,
and of them both have black hair
and wear glasses. How many of the students have black hair but do not
wear glasses?
A hiker is exploring a trail. The trail has three sections: the
first of the trail is along a
river, the next of the
trail is through a forest, and the remaining km of the trail is up a hill. How long
is the trail?
The operation is
defined by . The
value of
is
Part B: Each correct answer is worth 6.
Lauren plays basketball with her friends. She makes baskets. Each of these baskets is
worth either or points. Lauren scores a total of points. How many point baskets did she make?
Glen, Hao, Ioana, Julia, Karla, and Levi participated in the 2023
Canadian Team Mathematics Contest. On their team uniforms, each had a
different number chosen from the list , , , , , . Hao’s and Julia’s numbers were even.
Karla’s and Levi’s numbers were prime numbers. Glen’s number was a
perfect square. What was Ioana’s number?
Figure 1 shows an arrangement of lines with intersection point, and Figure 2 shows
an arrangement of lines with
intersection points.
What is the maximum number of intersection points that can appear in
an arrangement of lines?
The average (mean) of a list of numbers is . When one number is removed from the
list, the new average is . What
number was removed?
In , points
and lie on , as shown.
If , the measure
of is
The value of is
less than the value of .
The value of is less than the
value of .
Which of the following could be a value of ?
The first two hours of Melanie’s trip was spent travelling at
. The remaining
of Melanie’s trip
was spent travelling at . Melanie’s average speed during this trip is closest
to
A numerical value is assigned to each letter of the alphabet. The
value of a word is determined by adding up the numerical values of each
of its letters. The value of SET is , the value of HAT is , the value of TASTE is , and the value of MAT is . What is the value of the word
MATH?
In the diagram, has and
and rectangle has and .
The perimeter of
is equal to the perimeter of rectangle . What is the area of ?
If is a positive integer
between and , inclusive, what is the
maximum possible value for the sum of the digits of ?
Part C: Each correct answer is worth 8.
Each correct answer is an integer from 0 to 99, inclusive.
A table starts
with every entry equal to and is
modified using the following steps:
(i) adding to all three numbers
in any row;
(ii) adding to all three numbers
in any column.
After step (i) has been used a total of times and step (ii) has been used a
total of times, the table appears
as shown.
What is the value of ?
Pablo has solid that he assembles in
a larger cube.
If of the smaller cubes are red,
are blue, and are yellow, what is the smallest
possible surface area of the larger cube that is red?
A lock code is made up of four digits that satisfy the following
rules:
At least one digit is a ,
but neither the second digit nor the fourth digit is a .
Exactly one digit is a ,
but the first digit is not .
Exactly one digit is a .
The code includes a , or
the code includes a , or the code
includes two s.
How many codes are possible?
In the diagram, and
is the midpoint of . Two quarter circles are drawn: one
with centre and passing through
and , and the other with centre and passing through . Let be the area of the region that is
inside rectangle and outside
the larger quarter circle. Let be
the area that is inside the larger quarter circle, outside the smaller
quarter circle, and outside rectangle . Let be the positive difference between
and . What is the integer closest to ?
Each of , and is equal to a number from the list
, , , , , , , . There are triples with for which each of , and is equal to an integer.
What is the value of ?
Further Information
For students...
Thank you for writing the Pascal Contest!
Encourage your teacher to register you for the Fryer Contest which will be written in April.
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