2021 Galois Contest
(Grade 10)

April 2021
(in North America and South America)

April 2021
(outside of North American and South America)

©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
   • 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. The operation $\mathrm{△}$ is defined by $a\mathrm{△}b=a(2b+4)$ for integers $a$ and $b$. For example, $3\mathrm{△}6=3(2\times 6+4)=3(16)=48.$

   1. What is the value of $5\mathrm{△}1$?

   2. If $k\mathrm{△}2=24$, what is the value of $k$?

   3. Determine all values of $p$ for which $p\mathrm{△}3=3\mathrm{△}p$.

   4. Determine all values of $m$ for which $m\mathrm{△}(m+1)=0$.

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. How many ties did Team $P$ have at the end of the season?

   2. Team $Q$ 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. Explain why Team $R$ could not have finished the season with exactly $6$ ties.

   4. At 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. Diagonals $AC$ and $BD$ intersect at point $E$. What is the area of $\mathrm{△}ADE$?

   2. Point $P(0,p)$ lies on line segment $AB$. The area of trapezoid $BCDP$ is twice the area of $\mathrm{△}PAD$. What is the value of $p$?

   3. The 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 $5:3$.

4. 1. If ${\displaystyle \frac{5}{x}}+{\displaystyle \frac{14}{y}}=2$ and $x=6$, what is the value of $y$?

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

   3. Consider the equation ${\displaystyle \frac{16}{x}}+{\displaystyle \frac{25}{y}}=p$, 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

• 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