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2021 Galois Contest
(Grade 10)

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.

Questions

  1. The operation \(\triangle\) is defined by \(a \triangle b = a(2b +4)\) for integers \(a\) and \(b\). For example, \(3\triangle 6=3(2\times 6+4)=3(16)=48.\)

    1. LightbulbWhat is the value of \(5 \triangle 1\)?

    2. LightbulbIf \(k \triangle 2 = 24\), what is the value of \(k\)?

    3. Full solutionDetermine all values of \(p\) for which \(p \triangle 3 = 3\triangle p\).

    4. Full solutionDetermine all values of \(m\) for which \(m \triangle (m+1) = 0\).

  2. The organizer for a sports league with four teams has entered some of the end-of-season data into the table shown. Each team played \(27\) games and each game resulted in a win for one team and a loss for the other team, or in a tie for both teams. Each team earned 2 points for a win, 0 points for a loss, and 1 point for a tie.

    Team Name Games Played Number of Wins Number of Losses Number of Ties Total Points
    \(P\) 27 10 14 23
    \(Q\) 27
    \(R\) 27 25
    \(S\) 27

    1. LightbulbHow many ties did Team \(P\) have at the end of the season?

    2. LightbulbTeam \(Q\) had \(2\) more wins than Team \(P\) and \(4\) fewer losses than Team \(P\). How many total points did Team \(Q\) have at the end of the season?

    3. Full solutionExplain why Team \(R\) could not have finished the season with exactly \(6\) ties.

    4. Full solutionAt the end of the season, Team \(S\) had \(4\) more wins than losses. Show that Team \(S\) must have finished the season with a total of \(31\) points.

  3. Rectangle \(ABCD\) has vertices \(A(0,0)\), \(B(0, 12)\), \(C(6,12)\), and \(D(6,0)\).

    1. LightbulbDiagonals \(AC\) and \(BD\) intersect at point \(E\). What is the area of \(\triangle ADE\)?

    2. LightbulbPoint \(P(0,p)\) lies on line segment \(AB\). The area of trapezoid \(BCDP\) is twice the area of \(\triangle PAD\). What is the value of \(p\)?

    3. Full solutionThe line passing through \(U(0,u)\), \(V(2,4)\) and \(W(6,w)\) divides \(ABCD\) into two trapezoids. Determine all possible pairs of points \(U\) and \(W\) for which the ratio of the areas of these two trapezoids is \(5:3\).

    1. LightbulbIf \(\dfrac{5}{x} + \dfrac{14}{y} = 2\) and \(x=6\), what is the value of \(y\)?

    2. Full solutionDetermine all possible ordered pairs of positive integers \((x,y)\) that are solutions to the equation \(\dfrac{4}{x} +\dfrac{5}{y} = 1\).

    3. Full solutionConsider the equation \(\dfrac{16}{x} + \dfrac{25}{y} = p\), where \(p\) is a prime number and \(p\ge 5\). Determine all possible values of \(p\) for which there is at least one ordered pair of positive integers \((x,y)\) that is a solution to the equation.

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