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2021 Fryer Contest
(Grade 9)

April 2021
(in North America and South America)

April 2021
(outside of North American and South America)

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©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 Lightbulb
  2. FULL SOLUTION parts indicated by Full Solution

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

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:
  1. Please read the instructions for the contest.
  2. Write all answers in the answer booklet provided.
  3. For questions marked Lightbulb, place your answer in the appropriate box in the answer booklet and show your work.
  4. For questions marked Full Solution, 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.
  5. Diagrams are not drawn to scale. They are intended as aids only.
  6. 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 \(y=x^{3} -x\), you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

  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. LightbulbWhat is the area of each business card in \(\mbox{cm}^2\)?

    2. LightbulbSeveral business cards are printed without overlapping on a single \(20\) cm \(\times\) \(27\) cm page. If the entire page is used with no waste, how many business cards are printed?

    3. Full solutionSeveral business cards are to be printed on \(19\) cm \(\times\) \(29\) cm pages in one of two possible ways:

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

      • The landscape page layout is printed so that every card is positioned with its \(5\) cm edges parallel to the 29 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.

    Points O(0,0), F(240, 100), and G(240, 180) are plotted in the first quadrant of a Cartesian Plane. The vertical y-axis is labelled, 'Distance NOrth of School (m)', and the horizontal x-axis is labelled 'Distance East of School (m)'.

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

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

    2. LightbulbOn 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. Full solutionOn 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 \(2\leq n\leq 6\). 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. Lightbulb\(R_3\) is performed on the list \(5,2,3,1,4,6\). What is the new list?

    2. LightbulbA 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. Full solutionSuppose 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 \(\Box,\Box,\Box,\Box,\Box,3\)). 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. Full solutionDetermine 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 \(\Box,\Box,\Box,\Box,5,4\)).

  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.

    Four squares are lined up side-by-side so that they form a rectangle that is one square unit high and four squares unites long. The lower left corner (vertex) of the first square is labelled S, and the upper right corner (vertex) of the fourth square is labelled F. The indicated path starts at S and travels along the edges of squares as follows: right, up, right, down, right, right, up, and so ends at point F.

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

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

      Six squares are lined up side-by-side. The lower left vertex of the first square is labelled S, and the upper right vertex of the sixth square is labelled F. The upper right vertex of the second square is labelled X, and the lower right vertex of the sixth square is labelled Y.

    2. Full solution Explain why no \(SF\) path passes through all three of the vertices \(A\), \(B\) and \(C\) in the diagram shown.

      Four squares are lined up with two squares placed one on top of the other with the top left vertex labelled A and the bottom right vertex labelled B. One square is attached to the right of the top square with its top right vertex is labelled F. Another square is attached to the left of the bottom square with its top-left vertex labelled C and bottom-left vertex labelled S.

    3. Full solution Determine the number of \(SF\) paths in the diagram below.

      Fourty squares are lined up with two squares placed one on top of the other. Nineteen squares lined up side-by-side attach to the right of the top square with the top-right vertex of the right-most square labelled F. Nineteen squares lined up side-by-side attach to the left of the bottom square with the bottom-left vertex of the left-most square labelled S.

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

For teachers...

Visit our website cemc.uwaterloo.ca to