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2015 Galois Contest
(Grade 10)

Thursday, April 16, 2015
(in North America and South America)

Friday, April 17, 2015
(outside of North American and South America)

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©2015 University of Waterloo

Instructions

Time: \(75\) minutes

Number of Questions: 10

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. LightbulbIn the diagram, line 1 has equation \(y=2x+6\) and crosses the \(x\)-axis at \(P\).

      Point P lies on the negative x-axis, and point R lies on the positive x-axis.

      What is the \(x\)-intercept of line 1?

    2. Full solution Line 2 has slope \(-3\) and intersects line 1 at \(Q(3,12)\), as shown.

      Determine the equation of line 2.

    3. Full solutionLine 2 crosses the \(x\)-axis at \(R\), as shown.

      Determine the area of \(\triangle PQR\).

  2. On Wednesday, students at six different schools were asked whether or not they received a ride to school that day.

    1. LightbulbAt School A, there were 330 students who received a ride and 420 who did not. What percentage of the students at School A received a ride?

    2. LightbulbSchool B has 240 students, of whom \(30\%\) received a ride. How many more of the 240 students in School B needed to receive a ride so that \(50\%\) of the students in School B got a ride?

    3. Full solutionSchool C has 200 students, of whom \(45\%\) received a ride. School D has 300 students. When School C and School D are combined, the resulting group has \(57.6\%\) of students who received a ride. If \(x\%\) of the students at School D received a ride, determine \(x\).

    4. Full solutionSchool E has 200 students, of whom \(n\%\) received a ride. School F has 250 students, of whom \(2n\%\) received a ride. When School E and School F are combined, between \(55\%\) and \(60\%\) of the resulting group received a ride. If \(n\) is a positive integer, determine all possible values of \(n\).

    1. LightbulbIf \(n+5\) is an even integer, state whether the integer \(n\) is even or odd.

    2. Full solutionIf \(c\) and \(d\) are integers, explain why \(cd(c+d)\) is always an even integer.

    3. Full solutionDetermine the number of ordered pairs \((e,f)\) of positive integers where

      • \(e<f\),

      • \(e+f\) is odd, and

      • \(ef=300\).

    4. Full solutionDetermine the number of ordered pairs \((m,n)\) of positive integers such that \((m+1)(2n+m)=9000\).

  4. In the diagram, square \(BCDE\) has side length 2. Equilateral \(\triangle XYZ\) has side length 1. Vertex \(Z\) coincides with \(D\) and vertex \(X\) is on \(ED\).

    Vertex Y lies inside square BCDE.

    1. LightbulbWhat is the measure of \(\angle YXE\)?

    2. Full solution A move consists of rotating the square clockwise around a vertex of the triangle until a side of the square first meets a side of the triangle. The first move is a rotation about \(X\) and the second move is a rotation about \(Y\), as shown in the diagrams. (Note that the vertex of the triangle about which the square rotates remains in contact with the square during the rotation.)

      Square BCDE goes through the first move then the second move in the diagram. A description of the diagram is given in the following list.

      In subsequent moves, the square rotates about vertex \(Z\), then \(X\), then \(Y\), and so on. Determine, with justification, the total number of moves made from the beginning of the first move to when vertex \(D\) next coincides with a vertex of the triangle.

    3. Full solutionDetermine the length of the path travelled by point \(E\) from the beginning of the first move to when square \(BCDE\) first returns to its original position (that is, when \(D\) next coincides with \(Z\) and \(XZ\) lies along \(ED\)).

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