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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)

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©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 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.

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

    2. Lightbulb 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. Full Solution 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. Full Solution 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. Lightbulb Point \(E\) is on \(BC\), as shown.

      In rectangle ABCD, AD is the base of the rectangle and E is on the top side BC. ABCD is divided into 3 triangles by AE and DE. Triangle ADE is unshaded. Triangles BAE and CDE are shaded.

      What is the total area of the shaded regions?

    2. Lightbulb Point \(F\) is on \(BC\), and \(BD\) intersects \(AF\) at \(G\), as shown.

      In rectangle ABCD, AD is the base of the rectangle and point F is on the top side BC. BD is a diagonal that meets AF at G. Triangle BGA is shaded and the rest of the rectangle is unshaded.

      If the area of \(\triangle FGD\) is 5, what is the area of the shaded region?

    3. Full Solution 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.

      In rectangle ABCD, R is on the base AD and P is on the top side BC. AP and BR meet at S and form two shaded triangles ASR and BSP. RC and PD meet at Q and form two shaded triangles RQD and PQC. The rest of the rectangle is unshaded.

      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:

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

      • \(726\,194\)

      • \(726\,941\)

      • \(746\,291\)

      • \(627\,491\)

      • \(276\,491\)

      • \(926\,471\)

      • \(796\,421\)

      • \(726\,419\)

      • \(729\,461\)

      • \(716\,492\)

      • \(762\,491\)

      • \(726\,491\)

      • \(126\,497\)

      • \(721\,496\)

      • \(426\,791\)

      • \(724\,691\)

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

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

    3. Full Solution 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

For teachers...

Visit our website cemc.uwaterloo.ca to