2024 Cayley Contest
(Grade 10)
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 expression is equal to
If , the value of is
In , points
and are on and , respectively, such that and .
If , the
measure of is
At Wednesday’s basketball game, the Cayley Comets scored points.
At Friday’s game, they scored
as many points as they scored on Wednesday.
How many points did they score on Friday?
In the diagram, the two identical bases of the prism are shaped
like a star.
The area of each star-shaped base is . The depth of the prism (that is, the distance
between the star-shaped bases) is
cm. The volume of the prism is
Last year, Lloyd ate cookies in the percentages shown in the pie
chart.
The number of gingerbread cookies that he ate was two times the
number of sugar cookies that he ate. What percentage of the cookies that
he ate were gingerbread cookies?
If , the value of is
Which of the following integers is equal to a perfect square?
The sum of five consecutive odd integers is . The smallest of these integers
is
Two standard six-sided dice are rolled. What is the probability
that the product of the two numbers rolled is ?
Part B: Each correct answer is worth 6.
Arturo has an equal number of bills, of bills, and of bills. The total value of these
bills is . How many bills does Arturo have?
The Cayley Corner Store sells three types of toys: Exes, Wyes and
Zeds. All Exes are identical, all Wyes are identical, and all Zeds are
identical. The mass of Exes
equals the mass of Wyes. The
mass of Zed equals the mass of
Exes. The mass of Zed equals the mass of how many
Wyes?
In the diagram, quadrilateral has , , and . Also, and are perpendicular to .
The perimeter of quadrilateral is
Ten numbers have an average (mean) of . Two of those numbers are and . The average of the other eight
numbers is
A rectangle has width and
length , as shown in Figure 1. The
rectangle is cut along the horizontal and vertical dotted lines in
Figure 1 to produce four smaller rectangles as shown in Figure 2.
The sum of the perimeters of these four rectangles in Figure 2 is
. The value of is
Suppose that . (The expression
under the square root is the product of fractions.) The value of is
Each of the four digits of the integer is even. How many integers between
and , inclusive, have the property that
all four of their digits are even?
The line with equation is translated units to
the right. The equation of the resulting line is
In the diagram, is right-angled at .
Points , ,
are on , points , ,
are on , point is on , point is on , and point is on so that , and are squares.
The area of is and the area of is . The area of square is
Jiwei and Hari entered a race. Hari finished the race in of the time it took Jiwei to
finish. The next time that they raced the same distance, Jiwei increased
his average speed from the first race by %, while Hari maintained the same
average speed as in the first race. In this second race, Hari finished
the race in the same amount of time that it took Jiwei to finish. The
value of is
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 0 and is modified using the following
steps:
(i) adding 1 to all three numbers in any row;
(ii) adding 2 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 ?
For how many integers does
the line with the equation
intersect the line segment with endpoints and ?
Four semi-circles are arranged so that their diameters form a
by rectangle. A circle is drawn through
the four vertices of the rectangle. In the diagram, the region inside
the four semi-circles but outside the circle is shaded. The total area
of the shaded region is . What is
the integer closest to ?
A park has four paths, as shown in the map below.
It takes minutes to walk along
the path from to ,
minutes to walk along the path from to ,
minutes to walk along the path from to , and minutes to walk around the path that
begins and ends at . Rasheeqa goes
for a walk, starting at , walking
only in the directions indicated along the paths, never stopping to
rest, and finishing at .
If the walk takes a total of
minutes, how many possible values of are there with ?
Erin has an empty
grid consisting of
squares:
and follows the process below to construct a pattern:
(i) Place an X in any empty square.
(ii) If three or more consecutive squares each contain an X, stop and do
not add any more X’s; otherwise, go to step (i) and continue the
process.
For example, in a smaller grid, there are
different patterns that can be constructed:
(The last pattern may be obtained by placing X’s, in order, in
squares 1, 2, 4, and then 3.) By applying this process starting with the
empty grid, how many
different possible patterns can Erin construct?
Further Information
For students...
Thank you for writing the Cayley Contest!
Encourage your teacher to register you for Galois 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
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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