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2018 Fryer Contest
(Grade 9)

Thursday, April 12, 2018
(in North America and South America)

Friday, April 13, 2018
(outside of North American and South America)

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©2018 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.

Questions

  1. Sandy’s Fruit Market sells cherries, plums and blueberries. For each type of fruit, the price of one box is shown in the table.

    Fruit cherries plums blueberries
    Price $2.00 $3.00 $4.50

    1. LightbulbOn Monday, Shane visited Sandy’s Fruit Market. He bought 4 boxes of cherries, 3 boxes of plums, and 2 boxes of blueberries. How much did Shane pay in total?

    2. LightbulbOn Wednesday, Shane bought 2 boxes of plums. He bought some boxes of cherries, no blueberries, and spent $22.00 in total. How many boxes of cherries did he buy?

    3. Full solutionOn Saturday, Shane bought twice as many boxes of plums as boxes of cherries. He also bought 3 boxes of blueberries. How many boxes of cherries did Shane buy if he gave the cashier $100.00 and received $14.50 in change?

  2. In the diagrams shown, \(ABCD\) represents a rectangular field. There are three flagpoles: \(M\) on \(BC\), \(P\) on \(AD\), and \(Q\) on \(CD\). Paul runs along the path \(A \to D \to C \to M \to A\). Tyler runs along the path \(A \to P \to Q \to C \to B \to A\).

    Rectangle ABCE has the width AB marked as 105 meters. Side AD is marked as 200 meters, and flagpole M is placed on BC so that BM = 100 meters.Rectangle ABCDE has the width AB marked as 105 meters. Side BC is marked as 200 meters, flagpole P is placed on AD so that AP is 140 meters, and flagpole Q is placed on CD so that CD = 60 meters.

    1. LightbulbWhat is the length of \(MA\)?

    2. LightbulbWhat is the total distance that Tyler runs?

    3. Full solutionPaul and Tyler start running at the same time. Tyler runs at a speed of 145 m/min. Paul runs at a constant speed and finishes 1 minute after Tyler. Determine Paul’s speed, in m/min.

    1. LightbulbA line has equation \(y=2x-6\). What is its \(x\)-intercept and what is its \(y\)-intercept?

    2. LightbulbA line has equation \(y = kx - 6\), where \(k\neq 0\). What is its \(x\)-intercept? Express your answer in terms of \(k\).

    3. Full solutionA triangle is formed by the positive \(x\)-axis, the negative \(y\)-axis, and the line with equation \(y=kx-6\), where \(k >0\). The area of this triangle is 6. What is the value of \(k\)?

    4. Full solutionA triangle is formed by the positive \(x\)-axis, the line with equation \(y=mx-m^2\), and the line with equation \(y=2mx-m^2\). Determine all values of \(m>0\) for which the area of the triangle is \(\frac{54}{125}\).

  4. A Bauman number is a positive integer each of whose digits is 1 or 2. Each Bauman number consists of blocks of digits. Each block contains at least one digit and includes all of the consecutive equal digits. For example, 2222111112111 is a 13-digit Bauman number consisting of four blocks: a block of four 2s, then a block of five 1s, then a block of one 2, then a block of three 1s; 2222222 is a 7-digit Bauman number consisting of a single block of seven 2s.

    1. LightbulbHow many 3-digit Bauman numbers are there?

    2. LightbulbHow many 10-digit Bauman numbers consist of fewer than three blocks?

    3. Full solutionDetermine the number of Bauman numbers that consist of at most three blocks and have the property that the sum of the digits is 7.

    4. Full solutionSome Bauman numbers include a block of exactly 2018 2s. Determine the number of \(4037\)-digit Bauman numbers that include at least one block of exactly 2018 2s.

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