2021 Fryer Contest
(Grade 9)

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:

1. 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
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate locationin 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, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

1. A company sells rectangular business cards. Each card has dimensions $5\text{ cm}\times 9\text{ cm}$. Cards are printed on a page and then the page is cut to produce the individual cards.

   1. What is the area of each business card in ${\text{cm}}^2$?

   2. Several business cards are printed without overlapping on a single $20\text{ cm}\times 27\text{ cm}$ cm page. If the entire page is used with no waste, how many business cards are printed?

   3. Several business cards are to be printed on $19\text{ cm}\times 29\text{ cm}$ pages in one of two possible ways:

      • The portrait page layout is printed so that every card is positioned with its $5\text{ cm}$ edges parallel to the $19\text{ cm}$ edges of the page.

      • The landscape page layout is printed so that every card is positioned with its $5\text{ cm}$ edges parallel to the $29\text{ cm}$ edges of the page.

      Which of these two page layouts allows the greatest number of business cards from a single page?

      

2. Franklin and Giizhig travel from their school to their own homes each day. The school is located at $O(0,0)$. Franklin’s home is at $F(240,100)$ and Giizhig’s home is at $G(240,180)$. The straight paths from their school to each of their homes are shown on the graph.

   

   (Throughout this problem, all coordinates represent lengths in metres.)

   1. What is the distance, in metres, along the straight path from the school to Franklin’s home?

   2. On Monday, Franklin walks at a constant speed of $80$ m/min. How many minutes does it take Franklin to walk from school straight to his home?

   3. On Tuesday, Franklin and Giizhig leave school at the same time. Franklin walks at 80 m/min straight to his own home and then immediately turns and walks straight toward Giizhig’s home. Giizhig walks at $g$ m/min straight to her own home and then immediately turns and walks straight toward Franklin’s home. If they meet exactly halfway between their homes, what is the value of $g$?

3. Given a list of six numbers, the Reverse Operation, $R_n$, reverses the order of the first $n$ numbers in the list, where $n$ is an integer and $2\le n\le 6$. For example, if the list is $1,4,6,2,3,5$, then after performing $R_4$ the list becomes $2,6,4,1,3,5$.

   1. $R_3$ is performed on the list $5,2,3,1,4,6$. What is the new list?

   2. A Reverse Operation is performed on the list $1,2,3,4,5,6$. A second Reverse Operation is performed on the resulting list to give the final list $3,4,2,1,5,6$. Which two Reverse Operations were performed and in what order were they performed?

   3. Suppose that $m$ is the minimum number of Reverse Operations that need to be performed, in order, on the list $1,2,3,4,5,6$ so that 3 ends up in the last position (that is, the list takes the form $◻,◻,◻,◻,◻,3$). The value of $m$ can be determined by answering (i) and (ii), below.

      1. Find $m$ Reverse Operations and show that after performing them, the desired result is achieved (that 3 ends up in the last position).

      2. Explain why performing fewer than $m$ Reverse Operations can never achieve the desired result.

   4. Determine the minimum number of Reverse Operations that need to be performed, in order, on the list $1,2,3,4,5,6$ so that the last number in the list is 4 and the second last number in the list is 5 (that is, the list takes the form $◻,◻,◻,◻,5,4$).

4. An SF path starts at $S$, follows along the edges of the squares, never visits any vertex more than once, and finishes at $F$. An example of an $SF$ path is shown.

   

   (A vertex is a point where two or more of the squares’ edges meet.)

   1. In your solution booklet, draw the $SF$ path that passes through each vertex except $X$ and $Y$.

      

   2. Explain why no $SF$ path passes through all three of the vertices $A$, $B$ and $C$ in the diagram shown.

      

   3. Determine the number of $SF$ paths in the diagram below.

      

Further Information

For students...

Thank you for writing the Fryer 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