2025 Fryer Contest
(Grade 9)

Thursday, April 3, 2025
(in North America and South America)

Friday, April 4, 2025
(outside of North American and South America)

©2025 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.

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

1. Azizi sold chocolate bars from Monday to Friday over three weeks.

   1. During the first week, he sold the following numbers of chocolate bars:

      Monday	Tuesday	Wednesday	Thursday	Friday
      \(15\)	\(23\)	\(18\)	\(15\)	\(7\)

      How many chocolate bars did he sell in total during the first week?

   2. During the second week, he sold the following numbers of chocolate bars:

      Monday	Tuesday	Wednesday	Thursday	Friday
      \(7\)	\(15\)	\(x\)	\(23\)	\(x\)

      He sold a total of \(73\) chocolate bars during the second week. What is the value of \(x\)?

   3. During the third week, he sold \(y\) chocolate bars on Monday. Each day from Tuesday to Friday, he sold \(6\) more chocolate bars than he sold on the previous day. He sold a total of \(100\) chocolate bars during the third week. Determine the value of \(y\).

2. Three players, Ava, Beau, and Cato, are playing in a tournament. Each person plays exactly two games, one game against each of the other players. If a game ends in a tie, both players are awarded \(1\) point. Otherwise, the winning player is awarded \(W\) points, and the losing player is awarded \(0\) points. For example, if the tournament results are as shown below and \(W=2\), then points are awarded as follows:

   Game Results	Points Awarded
   	Ava	Beau	Cato
   Ava loses to Beau	\(0\)	\(2\)	——
   Beau and Cato tie	——	\(1\)	\(1\)
   Ava and Cato tie	\(1\)	——	\(1\)

   Suppose that \(S\) is equal to the sum of the points that have been awarded to the three players when the tournament has finished. In the example above, Ava is awarded \(1\) point, Beau is awarded \(3\) points, and Cato is awarded \(2\) points, so \(S=6\) in the example.

   1. Suppose the tournament results are as follows: Ava and Beau tie, Beau loses to Cato, Ava and Cato tie. If \(W=3\), what is the value of \(S\)?

   2. If \(W=4\) and \(S=6\), how many games ended in a tie?

   3. Suppose the tournament finishes with exactly one of the three games ending in a tie. If \(S=24\), what is the value of \(W\)?

   4. Suppose the tournament finishes with \(S=21\), but we are not told the results of the games. Determine all possible integer values of \(W\).

3. The prime factorization of \(784\) is \(2\times2\times2\times2\times7\times7\) or \(2^4\times7^2\), and so \(784\) is a perfect square because it can be written in the form \((2^2\times7)\times(2^2\times7)\). The prime factorization of \(45\) is \(3^2\times5\), and so \(45\) is not a perfect square. However, \(45\times5\) is a perfect square since \(45\times5=3^2\times5^2=(3\times5)\times(3\times5)\).

   1. What are all positive integers \(j\) with \(j \leq 20\) for which \(2^3\times 3^2\times j\) is a perfect square?

   2. Determine all positive integers \(k\) so that \(20\times k\) is both a perfect square and a divisor of \(3600\).

   3. Determine the number of ordered triples of positive integers \((a, b, c)\) so that \(a^2\times b^2\times c=2025\).

4. Parallelogram \(ABCD\) has vertices \(A(0,0)\), \(B(7,0)\), \(C(9, 4)\), and \(D(2, 4)\).

   1. Point \(E(4,4)\) lies on \(CD\). What is the sum of the areas of \(\triangle ABC\), \(\triangle ABD\) and \(\triangle ABE\)?

   2. Let \(G\) be a point with integer coordinates that lies on the perimeter of \(ABCD\). Suppose \(\triangle CDG\) has non-zero area. How many possibilities are there for the point \(G\)?

   3. Determine the sum of the areas of all triangles whose vertices all have integer coordinates and lie on the perimeter of \(ABCD\).

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