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2019 Fryer Contest
(Grade 9)

Wednesday, April 10, 2019
(in North America and South America)

Thursday, April 11, 2019
(outside of North American and South America)

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©2019 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

Useful Fact: It may be helpful to know that the sum of the \(n\) integers from \(1\) to \(n\) equals \(\frac{1}{2}n(n+1)\); that is, \(1 + 2 + 3 + ... + (n-1) + n = \frac{1}{2}n(n+1)\).

    1. LightbulbA rectangle with dimensions 7 by 8 is shown in Figure A. What is the perimeter of this figure?

    2. LightbulbA 3 by 1 rectangle is removed from one corner of a 7 by 8 rectangle, as shown in Figure B. What is the perimeter of this figure?

    3. Full solutionA 4 by 2 rectangle is removed from one corner of a \(k+4\) by \(k+2\) rectangle, as shown in Figure C. Suppose that the perimeter of Figure C is 56. Determine the value of the integer \(k\).

    4. Full solutionFour 4 by 7 rectangles are removed from the corners of a square having side length \(8n+1\), as shown in Figure D. Determine the largest integer \(n\) for which the perimeter of Figure D is less than 1000.

  2. Rope is fed into a machine at a constant rate of 2 metres per second. The machine can be set to cut off one piece of rope every \(t\) seconds for various values of \(t\). For example, if the machine is set to make one cut every 5 seconds, then 12 pieces of rope are cut off in 1 minute.

    1. LightbulbIf the machine is set to make one cut every 8 seconds, how many pieces of rope are cut off in 10 minutes?

    2. LightbulbIf the machine is set to make one cut every 3 seconds, what is the length of each piece of rope that is cut off?

    3. Full solutionIf each piece of rope that is cut off is 30 m long, determine the number of cuts per minute that the machine is set to make.

    4. Full solutionIf the machine is set to make 16 cuts per minute, determine the length of each piece of rope that is cut off.

  3. Tania lists the positive integers, in order, leaving out the integers that are multiples of 5. Her resulting list is \[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, \ldots.\]

    1. LightbulbHow many integers has Tania listed just before she leaves out the 6th multiple of 5?

    2. Full solutionIf the \(k\)th integer in Tania’s list is 2019, determine the value of \(k\).

    3. Full solutionDetermine the sum of the first 200 integers in Tania’s list.

  4. A Shonk sequence is a sequence of positive integers in which

    For example: \(2,6, 27\) is a Shonk sequence since \(6>2\) and \(27>6\) and \(2 \times 6 \times 27 = 324=18^2\).

    1. LightbulbIf \(12, x, 24\) is a Shonk sequence, what is the value of \(x\)?

    2. LightbulbIf \(28, y, z, 65\) is a Shonk sequence, what are the values of \(y\) and \(z\)?

    3. Full solutionDetermine the length of the longest Shonk sequence, each of whose terms is an integer between 1 and 12, inclusive. This means that your solution should include an example of a sequence of this longest length, as well as justification as to why no longer sequence is possible.

    4. Full solutionA sequence of four terms \(a,b,c,d\) is called a super-duper-Shonkolistic sequence (SDSS) exactly when each of \(a,b,c,d\) and \(a, b, c\) and \(b, c, d\) is a Shonk sequence. Determine the number of pairs \((m, n)\) such that \(m, 1176, n, 48\,400\) is an SDSS.

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