2017 Fryer Contest
(Grade 9)

Wednesday, April 12, 2017
(in North America and South America)

Thursday, April 13, 2017
(outside of North American and South America)

©2017 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. A store sells packages of red pens and packages of blue pens. Red pens are sold only in packages of 6 pens. Blue pens are sold only in packages of 9 pens.

   1. Igor bought 5 packages of red pens and 3 packages of blue pens. How many pens did he buy altogether?

   2. Robin bought 369 pens. She bought 21 packages of red pens. How many packages of blue pens did she buy?

   3. Explain why it is not possible for Susan to buy exactly 31 pens.

2. By finding a common denominator, we see that ${\displaystyle \frac{1}{3}}$ is greater than ${\displaystyle \frac{1}{7}}$ because ${\displaystyle \frac{7}{21}}>{\displaystyle \frac{3}{21}}$.
   Similarly, we see that ${\displaystyle \frac{1}{3}}$ is less than ${\displaystyle \frac{1}{2}}$ because ${\displaystyle \frac{2}{6}}<{\displaystyle \frac{3}{6}}$.

   1. Determine the integer $n$ so that ${\displaystyle \frac{n}{40}}$ is greater than ${\displaystyle \frac{1}{5}}$ and less than ${\displaystyle \frac{1}{4}}$.

   2. Determine all possible integers $m$ so that ${\displaystyle \frac{m}{8}}$ is greater than ${\displaystyle \frac{1}{3}}$ and ${\displaystyle \frac{m+1}{8}}$ is less than ${\displaystyle \frac{2}{3}}$.

   3. Fiona calculates her win ratio by dividing the number of games that she has won by the total number of games that she has played. At the start of a weekend, Fiona has played 30 games, has $w$ wins, and her win ratio is greater than 0.5. During the weekend, she plays five games and wins three of these games. At the end of the weekend, Fiona’s win ratio is less than 0.7. Determine all possible values of $w$.

3. When two chords intersect each other inside a circle, the products of the lengths of their segments are equal. That is, when chords $PQ$ and $RS$ intersect at $X$, $(PX)(QX)=(RX)(SX)$.

   

   1. In Figure A below, chords $DE$ and $FG$ intersect at $X$ so that $EX=8$, $FX=6$, and $GX=4$. What is the length of $DX$?

   2. In Figure B, chords $JK$ and $LM$ intersect at $X$ so that $JX=8y$, $KX=10$, $LX=16$, and $MX=y+9$. Determine the value of $y$.

   3. In Figure C, chord $ST$ intersects chords $PQ$ and $PR$ at $U$ and $V,$ respectively, so that $PU=m$, $QU=5$, $RV=8$, $SU=3$, $UV=PV=n$, and $TV=6$. Determine the values of $m$ and $n$.

      

4. Three students sit around a table. Each student has some number of candies. They share their candies using the following procedure:

   

   • Step 1: Each student with an odd number of candies discards one candy. Students with an even number of candies do nothing.

   • Step 2: Each student passes half of the candies that they had after Step 1 clockwise to the person beside them.

   • Step 1 and Step 2 are repeated until each of the three students has an equal number of candies. The procedure then ends.

   On Monday, Dave, Yona and Tam start with 3, 7 and 10 candies, respectively. After Step 1 and Step 2, the number of candies that each student has is given in the following table:

   	Dave	Yona	Tam
   Start	3	7	10
   After Step 1	2	6	10
   After Step 2	6	4	8

   1. When the procedure in the example above is completed, how many candies does each student have when the procedure ends?

   2. On Tuesday, Dave starts with 16 candies. Each of Yona and Tam starts with zero candies. How many candies does each student have when the procedure ends?

   3. On Wednesday, Dave starts with $2n$ candies. Each of Yona and Tam starts with $2n+3$ candies. Determine, with justification, the number of candies in terms of $n$ that each student has when the procedure ends.

   4. On Thursday, Dave starts with $2^{2017}$ candies. Each of Yona and Tam starts with zero candies. Determine, with justification, the number of candies that each student has when the procedure ends.

Further Information

For students...

Thank you for writing the Fryer 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