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2026 Fryer Contest
(Grade 9)

Wednesday, April 1, 2026
(in North America and South America)

Thursday, April 2, 2026
(outside of North American and South America)

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Âİ2026 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 Short answer
  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 Short answer, 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.

Questions

  1. In the grids below, dots are \(1\) unit apart horizontally and \(1\) unit apart vertically. Shapes are created by connecting dots with straight lines.

    1. Short answer In Figure 1, the shaded shape is a rectangle with width \(8\) and height \(6\) that has a \(1 \times 1\) square removed. What is the area of the shaded shape in Figure 1?

    2. Short answer In Figure 2, the shaded shape is a rectangle with width 8 and height \(6\) that has a triangle removed. What is the area of the shaded shape in Figure 2?

    3. Full solution In Figure 3, trapezoid \(ABCD\) has \(AD=5\), \(BC=3\) and \(CD=8\). Point \(G\) is placed vertically below \(C\) and point \(H\) is placed vertically below \(D\), so that \(GH\) is parallel to \(CD\) and the area of trapezoid \(ABGH\) is twice the area of \(ABCD\). Determine the length of \(BG\).

      A rectangle with horizontal width 8 and vertical height 6 with its bottom side modified as follows: Starting at the bottom-left vertex of the rectangle, move 3 units to the right, then 1 unit up, 1 unit right, and 1 unit down, then 4 more units to the right, ending at the bottom-right vertex.

      Figure 1

      A rectangle with width 8 and height 6 with its bottom side modified as follows: Starting at the bottom-left vertex, move 1 unit to the right, then diagonally to a dot 3 units right and 2 units up, then diagonally to a dot 3 units right and 2 units back down, then 1 more unit to the right, ending at the bottom-right vertex.

      Figure 2

      Trapezoid ABCD plotted in the top half of a grid of dots, with the height of the grid about double the height of the trapezoid. Starting at the top-left vertex A, vertex D is 5 units below A, vertex C is 8 units to the right of D, vertex B is 3 units above C, and vertex B connects diagonally to A.

      Figure 3

  2. List \(P\) contains all positive integers from \(1\) to \(2^{53}\), inclusive.

    1. Short answer How many numbers in list \(P\) can be written as \(2^k\) where \(k\) is a positive integer?

    2. Short answer Since \(4=2^2\), every power of \(4\) can be written as a power of \(2\). For example, \(4^3\) can be written as \(4^3=\left(2^2\right)^3=2^{2\times 3}=2^6\). In general, \(4^n=\left(2^2\right)^n=2^{2\times n}=2^{2n}\). How many numbers in list \(P\) can be written as \(4^\ell\) where \(\ell\) is a positive integer?

    3. Full solution Determine how many numbers in list \(P\) can be written as \(4^r\) where \(r\) is a positive integer, but cannot be written as \(8^t\) where \(t\) is a positive integer.

  3. An open-topped cylindrical container has points \(S\) and \(U\) placed diametrically opposite each other on the top edge. Points \(T\) and \(V\) on the bottom edge are vertically below \(S\) and \(U\) respectively, as shown in Figure 1. The cylinder is filled with water and then tipped so that some of the water spills out. The cylinder is then held in a stationary tipped position until the water stops spilling out and settles. At this time, the top surface of the water

    as shown in Figure 2.

    Figure 1

    Figure 2

    In the case when \(WV=0\), one half of the cylinder's volume contains water. In the questions that follow, each of the three cylinders has been filled with water, tipped, and is being held in a stationary tipped position.

    1. Short answer Suppose that \(WV=WU\). What fraction of the cylinder's volume contains water?

    2. Full solution Suppose that \(UV=1\) and \(WV=x\) where \(0\leq x < 1\). Determine an expression for the fraction of the cylinder's volume that contains water, written in terms of \(x\).

    3. Full solution Suppose that a cylinder with radius \(8 \text{ cm}\) and height \(12 \text{ cm}\) contains \(624\pi \text{ cm}^3\) of water when held in a stationary tipped position. Determine the length of \(WV\). Note: A cylinder with radius \(r\) and height \(h\) has volume \(\pi r^2h\).

  4. In a Dunbar sequence,

    For example, the first six terms of the Dunbar sequence with first term \(2\) and second term \(5\) are: \[2, 5, 7, 12, 19, 31\]

    1. Short answer If the fifth term in a Dunbar sequence is \(57\) and the third term is \(20\), what is the first term in the sequence?

    2. Full solution Suppose that the first and second terms in a Dunbar sequence are \(a\) and \(b\) respectively. Determine all possible pairs \((a,b)\) for which the sixth term of the sequence is equal to \(104\).

    3. Full solution Suppose that the first and second terms in another Dunbar sequence are \(c\) and \(d\) respectively. Determine all possible pairs \((c,d)\) for which the product of the seventh term and the eighth term is equal to \(41\,440\).

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