2023 Hypatia Contest
(Grade 11)

Wednesday, April 5, 2023
(in North America and South America)

Thursday, April 6, 2023
(outside of North American and South America)

©2023 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

1. A game is played in which each throw of a ball lands in one of two holes: the closer hole or the farther hole. A throw landing in the closer hole scores 2 points, while a throw landing in the farther hole scores 5 points. A player’s total score is equal to the sum of the scores on their throws.

   1. Jasmin had 3 throws that each scored 2 points and 4 throws that each scored 5 points. What was Jasmin’s total score?

   2. Sam had twice as many throws that scored 2 points as throws that scored 5 points. If Sam’s total score was $36$ points, how many throws did Sam take?

   3. Tia had $t$ throws that each scored 2 points and $f$ throws that each scored 5 points. If Tia’s total score was 37 points, determine all possible ordered pairs $(t,f)$.

   4. The game is changed so that each throw scores 6 or 21 points instead of $2$ or $5$. Explain whether or not it is possible to have a total score of $182$ points.

2. In each question below, $ABCD$ is a rectangle with $AB=2$ and $AD=15$.

   1. Point $E$ is on $BC$, as shown.

      

      What is the total area of the shaded regions?

   2. Point $F$ is on $BC$, and $BD$ intersects $AF$ at $G$, as shown.

      

      If the area of $\mathrm{△}FGD$ is 5, what is the area of the shaded region?

   3. Point $P$ is on $BC$ and $R$ is on $AD$. $BR$ and $AP$ intersect at $S$ and $PD$ and $RC$ intersect at $Q$, as shown.

      

      If the area of $PQRS$ is $6$, determine the total area of the shaded regions.

3. For any positive integer with three or more different, non-zero digits, let a cousin be defined as the result of switching two digits of the integer. For example, the integer $156$ has three cousins:

   • $516$ (obtained by switching the 1st and 2nd digits),

   • $651$ (obtained by switching the 1st and 3rd digits), and

   • $165$ (obtained by switching the 2nd and 3rd digits).

   1. In no particular order, five of the six cousins of 6238 are listed below. Which cousin is missing from the list?

      • 2638

      • 6328

      • 3268

      • 6283

      • 8236

   2. In no particular order, the following list contains an original integer as well as all of its cousins. What is the original integer?

      • $726194$

      • $726941$

      • $746291$

      • $627491$

      • $276491$

      • $926471$

      • $796421$

      • $726419$

      • $729461$

      • $716492$

      • $762491$

      • $726491$

      • $126497$

      • $721496$

      • $426791$

      • $724691$

   3. Suppose that $c$ and $d$ are distinct, non-zero digits. The three-digit integer $cd3$ minus one of its cousins is equal to the three-digit integer $d95$. Determine the values of $c$ and $d$ and show that no other values are possible.

   4. Suppose that $m$ and $n$ are distinct, non-zero digits. The sum of the six cousins of the four-digit integer $mn97$ is equal to the five-digit integer $nmnm7$. Determine the values of $m$ and $n$ and show that no other values are possible.

4. The Great Math Company has a random integer generator which produces an integer from $1$ to $9$ inclusive, where each integer is generated with equal probability. Each member of the Multiplication Team uses this generator a certain number of times and then calculates the product of their integers.

   1. Amarpreet uses the generator 3 times. What is the probability that the product is a prime number?

   2. Braxton uses the generator 4 times. Determine the probability that the product is divisible by 5, but not divisible by 7.

   3. Camille uses the generator 2023 times. Let $p$ be the probability that the product is not divisible by 6. Determine the ones digit of the integer equal to $p\times 9^{2023}$.

Further Information

For students...

Thank you for writing the Hypatia 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