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2024 Galois Contest
(Grade 10)

Thursday, April 4, 2024
(in North America and South America)

Friday, April 5, 2024
(outside of North American and South America)

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©2024 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.

Questions

  1. Three students are helping to expand their school’s garden. Initially, the garden has a length of \(5\text{ m}\) and a width of \(4\text{ m}\), as shown.

    The rectangular garden with dimensions shown.

    1. Lightbulb Rob adds two additional \(2\text{ m}\) by \(4\text{ m}\) plots side by side next to the initial garden, as shown.

      The initial garden and two additional plots are placed side by side, with their 4 m sides touching, forming a rectangle with width 4 m and total length of 5 m plus 2 m plus 2 m.

      What is the total area of the expanded garden after Rob adds these two plots?

    2. Lightbulb Kirima adds a path around three sides of the previous garden, as shown.

      A straight path that travels along one of the longer sides of the larger rectangle, then turns and travels along one of the shorter sides, then turns and travels back along the other longer side.

      If the width of the path is \(1\text{ m}\), what is the total combined area of the garden and the path?

    3. Full solution Noah adds \(n\) additional \(2\text{ m}\) by \(4\text{ m}\) plots to the previous version of the garden (in part (b)), and then continues the \(1\text{ m}\) wide path so that it surrounds the entire garden, as shown. If the total combined area of the garden and the path is \(150 \text{m}^2\), determine the value of \(n\).

  2. When a point \((x,y)\) is rotated \(90\degree\) clockwise about the origin, the resulting coordinates are \((y, -x)\). We call this rotation \(R\). When a point \((x,y)\) is translated up \(2\) units, the resulting coordinates are \((x,y+2)\). We call this translation \(T\). For example, beginning with the point \((8,-2)\), and applying \(R\) then \(T\), the resulting coordinates are \((-2,-6)\), as shown: \[(8,-2) \overset{R}{\longrightarrow} (-2,-8) \overset{T}{\longrightarrow} (-2,-6)\]

    1. Lightbulb Beginning with the point \((5,11)\), and applying \(R\) then \(T\), what are the resulting coordinates?

    2. Lightbulb Beginning with the point \((-3,7)\), when \(R\) is applied \(5\) times, what are the resulting coordinates?

    3. Full solution Consider the following sequence of transformations: \(R\), then \(R\) again, and then \(T\). Beginning with the point \((9,1)\), this sequence \(R,R,T\) is repeated a total of \(11\) times. Determine the resulting coordinates.

  3. Seven black balls numbered \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), and \(7\), are placed in a hat. Balls are drawn randomly one at a time from the hat. When a ball is drawn, it is neither replaced by another ball nor returned to the hat.

    1. Lightbulb What is the probability that the first ball drawn is even-numbered?

    2. Lightbulb What is the probability that the sum of the numbers on the first two balls drawn is equal to \(5\)?

    3. Full solution Determine the probability that the sum of the numbers on the first two balls drawn is greater than or equal to \(6\).

    4. Full solution An eighth ball is added to the hat. This eighth ball is gold and it is numbered with an integer \(k\), where \(1\leq k\leq 7\). The probability that the sum of the numbers on the first two balls drawn is greater than or equal to 7 is \(\frac{3}{4}\). Determine the value of \(k\).

  4. In a \(3\times n\) rectangular grid, two cells are neighbours if they share an edge. A \(3\times n\) Griffin Grid is a \(3\times n\) grid, with \(n\geq2\), having the following properties:

    An example of a \(3\times 2\) Griffin Grid is shown.

    \(-1\) \(1\)
    \(-1\) \(-1\)
    \(-1\) \(1\)

    1. Lightbulb Fill in the empty cells of the \(3 \times 5\) grid shown so that it is a Griffin Grid.

      \(-1\)
      \(1\)
      \(-1\)

    2. Full solution Determine the total number of \(3\times 5\) Griffin Grids.

    3. Full solution Let \(S\) be the sum of the numbers of \(3\times n\) Griffin Grids for \(2\leq n\leq 2024\). Determine the value of \(S\).

Further Information

For students...

Thank you for writing the Galois 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