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2024 Fryer Contest
(Grade 9)

Thursday, April 4, 2024
(in North America and South America)

Friday, April 5, 2024
(outside of North American and South America)

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©2024 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.

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, \(1+2+3+\cdots+ (n-1)+n=\frac{1}{2}n(n+1)\).

Questions

  1. In a sequence of integers, the 1st term is \(3\). Each new term is obtained by adding \(6\) to the previous term. In this sequence, the first four terms are \(3\), \(9\), \(15\), \(21\).

    1. Lightbulb What is the 5th term?

    2. Lightbulb What is the average (mean) of the 4th, 5th and 6th terms?

    3. Lightbulb What is the 20th term?

    4. Full solution Determine the smallest term that is greater than \(1000\).

  2. Each day, Ella loaded her truck with red shirts and blue shirts for delivery to two stores. At Store 1, she dropped off some shirts, and at Store 2 she dropped off all remaining shirts.

    1. Lightbulb On Monday, she loaded \(800\) red shirts and \(200\) blue shirts. At Store 1, she dropped off only red shirts. At Store 2, \(50\%\) of the shirts dropped off were red. How many red shirts were dropped off at Store 1?

    2. Lightbulb On Tuesday, she loaded \(5x\) red shirts and \(5x\) blue shirts. At Store 1, she dropped off \(40\%\) of the red shirts, and no blue shirts. What percentage of the shirts dropped off at Store 2 were blue?

    3. Full solution On Wednesday, she loaded \(3y\) red shirts and \(y\) blue shirts. At Store 1, she dropped off some red shirts and no blue shirts. Additionally, she picked up some green shirts. At Store 2, she dropped off all remaining shirts and the numbers of red, green, and blue shirts dropped off were equal. Of all the shirts that she dropped off on Wednesday, determine the percentage that were green.

  3. A square slice of bread, \(ABCD\), has dimensions \(30 \times 30\). The slice of bread has crusts on three of its edges, as shown by the bolded sides \(AB\), \(BC\) and \(CD\) in Figure 1.

    Figure 1

    When the slice is cut into smaller pieces, the smaller pieces are called fair if

    1. Lightbulb In Figure 2, \(M\) is the midpoint of \(AD\) and \(MN\) (with \(N\) inside square \(ABCD\)) is perpendicular to \(AD\).

      Figure 2

      If \(ABCD\) is cut along the lines \(MN\), \(NB\) and \(NC\) to create three fair pieces, what is the length of \(MN\)?

    2. Lightbulb In Figure 3, \(M\) is the midpoint of \(AD\), \(MT\) (with \(T\) inside square \(ABCD\)) is perpendicular to \(AD\), and \(S\) lies on \(MT\).

      Point U is on AB, point V is on CD, and points P and Q are on BC with P closer to B and Q closer to C.

      Figure 3

      If \(ABCD\) is cut along the lines \(MT\), \(TP\), \(TQ\), \(SU\), and \(SV\) to create five fair pieces, what is the length of \(PQ\)?

    3. Full solution In Figure 3, determine the length of \(ST\).

  4. A game is played with spinners having three equally sized sections numbered with three distinct integers. Each game is played between two players. Each player spins their spinner once and the player who spins the highest number is the winner. Using either of the two spinners shown, a player could spin a \(2\), a \(4\) or a \(7\), each with a probability of \(\frac{1}{3}\), and so the spinners would be considered to be the same and both are labelled \(\{2,4,7\}\).

    A circular spinner with three equally sized sections. Moving clockwise around the spinner, the sections are labelled 2, 4, then 7. A circular spinner with three equally sized sections. Moving clockwise around the spinner, the sections are labelled 2, 7, then 4.

    1. Lightbulb Alice spins the spinner \(\{5, 9, 11\}\) and Binh spins the spinner \(\{1, 8, 10\}\). What is the probability that Alice wins?

    2. Full solution Carole makes the spinner \(\{1, 5, 10\}\). Darsh makes a spinner by choosing three distinct integers from \(2, 3, 4, 6, 7, 8, 9\). Determine how many different spinners Darsh can make so that his probability of winning is greater than Carole’s probability of winning.

    3. Full solution Erin makes the spinner \(\{5, 8, 15\}\) and Fynn makes the spinner \(\{2, 10, 18\}\). Gina makes a spinner \(\{x,y,z\}\) by choosing three integers \(x<y<z\) from the list \[1,3, 4, 6, 7,9,11,12,13,14,16,17,19,20\] Suppose that

      • the probability that Fynn beats Erin is equal to \(p\),

      • the probability that Erin beats Gina is equal to \(q\), and

      • the probability that Gina beats Fynn is equal to \(r\).

      Determine how many different spinners Gina can make so that \(p=q=r\).

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