2022 Fryer Contest
(Grade 9)

Tuesday, April 12, 2022
(in North America and South America)

Wednesday, April 13, 2022
(outside of North American and South America)

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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:

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
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, $π + 1$ and $1 - \sqrt{2}$ are simplified exact numbers.

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:

Please read the instructions for the contest.
Write all answers in the answer booklet provided.
For questions marked , place your answer in the appropriate box in the answer booklet and show your work.
For questions marked , 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.
Diagrams are not drawn to scale. They are intended as aids only.
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

In a game, a player throws a ball at a target. If they hit the target, then 7 points are added to their score. If they miss the target, then $3$ points are subtracted from their score. A player’s score begins at 0, and it is possible for a player to have a negative score.
1. What is Shane’s score after $6$ throws if $4$ of the throws are hits and 2 of the throws are misses?
2. After exactly $h$ hits and 6 misses, Susan’s score is 59. What is the value of $h$ ?
3. After exactly $20$ throws, Souresh’s score is greater than 85 and less than 105. If exactly $m$ of these throws are misses, determine all possible values of the positive integer $m$ .
1. Two identical rectangles, $A B C D$ and $E F G H$ , each with area 13 cm $^{2}$ , overlap as shown.
  
  The area of the overlapped region, rectangle $E F C D$ , is 5 cm $^{2}$ . What is the area of rectangle $A B G H$ ?
2. Two identical right-angled triangles, $J K L$ and $M L K$ , overlap along side $K L$ , as shown.
  
  Sides $J L$ and $M K$ intersect at $N$ . The area of the overlapped region, $△ K L N$ , is equal to half of the area of $△ J K L$ . The area of the figure $J K L M N$ is $48 {cm}^{2}$ . If $J K = 6$ cm, determine the length of $K L$ .
3. Rectangle $P Q R S$ and $△ P Q T$ overlap so that $R$ lies on $Q T$ , and $R S$ intersects $P T$ at $U$ , as shown.
  
  The area of rectangle $P Q R S$ is 108 cm $^{2}$ , and the area of $△ P Q T$ is 81 cm $^{2}$ . If the area of the figure $P Q T U S$ is 117 cm $^{2}$ , determine the area of the overlapped region, $P Q R U$ .
If an integer $n$ is written as a product of prime numbers, this product (known as its prime factorization) can be used to determine the number of positive factors of $n$ . For example, the prime factorization of $28 = 2 \times 2 \times 7 = 2^{2} \times 7^{1}$ . The positive factors of 28 are: $\begin{aligned} 28 & = 2^{2} \times 7^{1} \\ 14 & = 2^{1} \times 7^{1} \\ 7 & = 2^{0} \times 7^{1} \\ 4 & = 2^{2} \times 7^{0} \\ 2 & = 2^{1} \times 7^{0} \\ 1 & = 2^{0} \times 7^{0} \end{aligned}$ Each positive factor includes $2$ , $1$ or $0$ twos, $1$ or $0$ sevens, and no other prime numbers. Since there are 3 choices for the number of twos, and 2 choices for the number of sevens, there are $3 \times 2 = 6$ positive factors of $28$ .
1. How many positive factors does $675$ have?
2. A positive integer $n$ has the positive factors $9$ , $11$ , $15$ , and $25$ and exactly fourteen other positive factors. Determine the value of $n$ .
3. Determine the number of positive integers less than $500$ that have the positive factors $2$ and $9$ and exactly ten other positive factors.
Franco and Sarah play a game four times using the following rules:
- (R1) The game starts with two jars, each of which might contain some beans.
- (R2)Franco goes first, Sarah goes second and they continue to alternate turns.
- (R3) On each turn, the player removes a pre-determined number of beans from one of the jars. If neither jar has enough beans in it, the player cannot take their turn and loses. If only one jar has enough beans in it, the player must remove beans from that jar. If both jars have enough beans, the player chooses one of the jars and removes the beans from that jar.
- (R4) Franco must attempt to remove 1 bean on his first turn, 3 beans on his second turn, and 4 beans on his third turn. On each of his following sets of three turns, Franco must continue to attempt to remove 1, 3 and 4 beans in sequence.
- (R5) Sarah must attempt to remove 2 beans on her first turn and 5 beans on her second turn. On each of her following sets of two turns, Sarah must continue to attempt to remove 2 and 5 beans in sequence.
- (R6) A player is declared the winner if the other player loses, as described in (R3).
For example, if the game begins with 10 beans in one jar and 10 beans in the other jar, the sequence of play could be:

Turn Number 1 2 3 4 5 6 7

Number of beans removed by Franco 1 3 4 1

Number of beans removed by Sarah 2 5 2

Number of beans remaining in the jars $10, 9$ $10, 7$ $7, 7$ $7, 2$ $3, 2$ $1, 2$ $0, 2$

On the next turn, Sarah cannot remove 5 beans since the greatest number of beans remaining in either jar is 2 and so after exactly 7 turns, Sarah loses and Franco wins.
1. At the beginning of the first game, there are 40 beans in one jar and 0 beans in the other jar. After a total of 10 turns (5 turns for each of Franco and Sarah), what is the total number of beans left in the two jars?
2. At the beginning of the second game, there are 384 beans in one jar and 0 beans in the other jar. The game ends with a winner after a total of exactly $n$ turns. What is the value of $n$ ?
3. At the beginning of the third game, there are 17 beans in one jar and 6 beans in the other jar. There is a winning strategy that one player can follow to guarantee that they are the winner. Determine which player has a winning strategy and describe this strategy. (A winning strategy is a way for a player to choose a jar on each turn so that they win no matter the choices of the other player.)
4. At the beginning of the fourth game, there are 2023 beans in one jar and 2022 beans in the other jar. Determine which player has a winning strategy and describe this strategy.

Turn Number	1	2	3	4	5	6	7
Number of beans removed by Franco	1		3		4		1
Number of beans removed by Sarah		2		5		2
Number of beans remaining in the jars	$10, 9$	$10, 7$	$7, 7$	$7, 2$	$3, 2$	$1, 2$	$0, 2$

Further Information

For students...

Thank you for writing the Fryer Contest!

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