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)

©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
   • worth 2 or 3 marks each
   • full marks are given for a correct answer which is placed in the box
   • part marks are awarded if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate location in the answer booklet
   • marks awarded for completeness, clarity, and style of presentation
   • a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

• Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages.
• Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

Questions

1. 1. In the diagram, line 1 has equation $y=2x+6$ and crosses the $x$-axis at $P$.

      

      What is the $x$-intercept of line 1?

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

      Determine the equation of line 2.

   3. Line 2 crosses the $x$-axis at $R$, as shown.

      Determine the area of $\mathrm{△}PQR$.

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

   1. At 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. School B has 240 students, of whom $30\mathrm{\%}$ received a ride. How many more of the 240 students in School B needed to receive a ride so that $50\mathrm{\%}$ of the students in School B got a ride?

   3. School C has 200 students, of whom $45\mathrm{\%}$ received a ride. School D has 300 students. When School C and School D are combined, the resulting group has $57.6\mathrm{\%}$ of students who received a ride. If $x\mathrm{\%}$ of the students at School D received a ride, determine $x$.

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

3. 1. If $n+5$ is an even integer, state whether the integer $n$ is even or odd.

   2. If $c$ and $d$ are integers, explain why $cd(c+d)$ is always an even integer.

   3. Determine the number of ordered pairs $(e,f)$ of positive integers where

      • $e<f$,

      • $e+f$ is odd, and

      • $ef=300$.

   4. Determine 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 $\mathrm{△}XYZ$ has side length 1. Vertex $Z$ coincides with $D$ and vertex $X$ is on $ED$.

   

   1. What is the measure of $\mathrm{\angle }YXE$?

   2. 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.)

      

      Hide/Reveal Description of the Diagram
      • 'Beginning of First Move' is the original position of square BCDE and triangle XYZ as already described.
      • 'During First Move' shows X still on ED, but now both Y and Z lie inside the square.
      • 'End of First Move' shows X still on ED and Z still inside the square, but Y now coincides with E.
      • 'End of Second Move' shows Y coinciding with E, Z lying on EB, and X lying inside the square.

      ---

      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. Determine 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

• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
• Information about careers in and applications of mathematics and computer science

For teachers...

Visit our website cemc.uwaterloo.ca to

• Obtain information about future contests
• Look at our free online courseware for high school students
• Learn about our face-to-face workshops and our web resources
• Subscribe to our free Problem of the Week
• Investigate our online Master of Mathematics for Teachers
• Find your school's contest results