CEMC Banner

2015 Fryer Contest
(Grade 9)

Thursday, April 16, 2015
(in North America and South America)

Friday, April 17, 2015
(outside of North American and South America)

University of Waterloo Logo

©2015 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.

  1. A company builds cylinders. Its Model A cylinder has radius \(r=10\) cm and height \(h=16\) cm.


    Volume of a Cylinder: \(V=\pi r^2h\)

    1. Lightbulb What is the volume in cm\(^3\) of a Model A cylinder?

    2. LightbulbThe company also builds a Model B cylinder having a radius of 8 cm. Each Model B cylinder has the same volume as each Model A cylinder. What is the height in cm of a Model B cylinder?

    3. Full solutionThe company makes a rectangular box, called Box A, that holds six Model A cylinders. The cylinders are placed into the box vertically and tightly packed, as shown.

      The cylinders are arranged in the box in two rows of three. The height of the box is exactly the same as the height of each cylinder.

      Determine the volume in cm\(^3\) of Box A.

    4. LightbulbThe company makes another rectangular box, called Box B, that holds six Model B cylinders. The cylinders are placed into the box vertically and tightly packed, just as was shown in part (c). State whether the volume of Box B is less than, greater than, or equal to, the volume of Box A.

  2. In Canada, a quarter is worth $0.25, a dime is worth $0.10, and a nickel is worth $0.05.

    1. LightbulbSusan has 3 quarters, 18 dimes and 25 nickels. What is the total value of Susan’s coins?

    2. Lightbulb Allen has equal numbers of dimes and nickels, and no other coins. His coins have a total value of $1.50. How many nickels does Allen have?

    3. Full solution Elise has $10.65 in quarters and dimes. If Elise has \(x\) quarters and \(2x+3\) dimes, what is the value of \(x\)?

  3. A formula for the sum of the first \(n\) positive integers is \(1+2+3+\cdots+n=\dfrac{n(n+1)}{2}\).
    For example, to calculate the sum of the first 4 positive integers, we evaluate \(1+2+3+4=\dfrac{4(4+1)}{2}=10\).

    1. LightbulbWhat is the sum of the first 200 positive integers, \[1+2+3+\cdots+198+199+200~?\]

    2. Full solutionCalculate the sum of the 50 consecutive integers beginning at 151, that is, \[151+152+153+\cdots+198+199+200~.\]

    3. Full solutionStarting with the sum of the first 1000 positive integers, \(1+2+3+\cdots+999+1000\), every third integer is removed to create the new sum \[1+2+4+5+7+8+10+11+\cdots+998+1000~.\] Calculate this new sum.

  4. The token \(\bullet\) is placed on a hexagonal grid, as shown.

    An alternative format for the hexagonal grid follows.

    At each step, the token can be moved to an adjacent hexagon in one of the three directions \(\nwarrow, \uparrow, \nearrow\).
    (The token can never be moved in any of the three directions, \(\swarrow, \downarrow, \searrow\).)

    1. LightbulbWhat is the minimum number of steps required to get the token to the hexagon labelled \(A\)?

    2. Full solutionWith justification, determine the maximum number of steps that can be taken so that the token ends at \(A\).

    3. Full solutionUsing exactly 5 steps, the token can end at the hexagon labelled \(C\) in exactly 20 different ways. Using exactly 5 steps, the token can end at \(n\) different hexagons in at least 20 different ways. Determine, with justification, the value of \(n\).

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