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)

©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
   • worth 2 or 3 marks each
   • full marks are given for a correct answer which is placed in the box
   • part marks are awarded if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate location in the answer booklet
   • marks awarded for completeness, clarity, and style of presentation
   • a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

• Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages.
• Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

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

   
   Volume of a Cylinder: $V=\pi r^{2}h$

   1. What is the volume in cm${}^3$ of a Model A cylinder?
      

   2. The 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. The 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.

      

      Determine the volume in cm${}^3$ of Box A.

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

   2. 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. 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 $1+2+3+\cdots +n={\displaystyle \frac{n(n+1)}{2}}$.
   For example, to calculate the sum of the first 4 positive integers, we evaluate $1+2+3+4={\displaystyle \frac{4(4+1)}{2}}=10$.

   1. What is the sum of the first 200 positive integers, $$1+2+3+\cdots +198+199+200?$$

   2. Calculate the sum of the 50 consecutive integers beginning at 151, that is, $$151+152+153+\cdots +198+199+200.$$

   3. Starting 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.

   

   Hide/Reveal Alternative Format for Hexagonal Grid
   The grid is made of 14 columns of tightly-packed hexagons, each sharing adjacent sides with no gaps between. Each column has 6 hexagons. The token is placed in the hexagon that is at the very bottom of the ninth column (numbering left to right). Hexagon A is at the very top of the fourth column. Hexagon C is the third down of the eleventh column. Note that the token may only be moved upward, either straight up, north-west, or north-east, and never downward (never straight down, never south-west, never south-east).

   ---

   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. What is the minimum number of steps required to get the token to the hexagon labelled $A$?

   2. With justification, determine the maximum number of steps that can be taken so that the token ends at $A$.

   3. Using 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

• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
• Information about careers in and applications of mathematics and computer science

For teachers...

Visit our website cemc.uwaterloo.ca to

• Obtain information about future contests
• Look at our free online courseware for high school students
• Learn about our face-to-face workshops and our web resources
• Subscribe to our free Problem of the Week
• Investigate our online Master of Mathematics for Teachers
• Find your school's contest results