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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)

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©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 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.
  7. No student may write more than one of the Fryer, Galois, and Hypatia Contests in the same year.

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

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

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

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

    1. LightbulbDetermine the integer \(n\) so that \(\dfrac{n}{40}\) is greater than \(\dfrac{1}{5}\) and less than \(\dfrac{1}{4}\).

    2. LightbulbDetermine all possible integers \(m\) so that \(\dfrac{m}{8}\) is greater than \(\dfrac{1}{3}\) and \(\dfrac{m+1}{8}\) is less than \(\dfrac{2}{3}\).

    3. Full solution 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 \(X\), \((PX)(QX)=(RX)(SX)\).

    Points P, S, Q, R lie on the circumference of a circle so that PQ intersects RS at the point X, which lies inside the circle. Lengths are marked as follows: PX=3, QX=4, RX=6, and SX=2.

    1. LightbulbIn 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. Full solutionIn 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. Full solutionIn 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:

    Dave, Tam, and Yona around a circular. Clockwise arrows point from Dave to Yona, from Yona to Tam, and from Tam to Dave.

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

    2. LightbulbOn 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. Full solutionOn 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. Full solutionOn 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

For teachers...

Visit our website cemc.uwaterloo.ca to