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2025 Hypatia Contest
(Grade 11)

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

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

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©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 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. To begin, tokens are distributed unequally between two people. Every minute following this, each person receives \(2\) more tokens.

    1. Lightbulb Tokens are distributed between Abi and Brody with Abi receiving \(30\) tokens and Brody receiving \(10\) tokens. How many tokens does Abi have in total after \(7\) minutes?

    2. Lightbulb A total of \(80\) tokens are distributed between Carl and Desiree. After \(12\) minutes, Desiree has \(37\) tokens. How many tokens did Carl start with?

    3. Full Solution A total of \(100\) tokens are distributed between Essi and Francis, with Essi receiving \(12\) tokens. After \(t\) minutes, Francis has \(3\) times as many tokens as Essi. Determine the value of \(t\).

    1. Lightbulb A rectangle, \(\mathcal{R}_1\), has length \(5 \text{ cm}\) and width \(4 \text{ cm}\). The length of the rectangle is increased by \(10\%\) and its width remains unchanged. What is the area of the resulting rectangle?

    2. Full Solution The area of a square is \(100 \text{ cm}^2\). When its length is increased by \(30\%\) and its width is decreased by \(30\%\), the area of the resulting rectangle is less than \(100 \text{ cm}^2\). Determine the percentage by which the area decreased.

    3. Full Solution The length of a rectangle, \(\mathcal{R}_2\), is increased by \(x\%\) and its width is decreased by \(20\%\). If the area of the resulting rectangle is equal to the area of the original rectangle, determine the value of \(x\).

  3. A parabola with equation \(y = ax^2 + bx + c\), where \(a=1\) or \(a=-1\), and a line intersect at points \(P(p, q)\) and \(R(r, s)\) for some real numbers \(p\), \(q\), \(r\), and \(s\). In this case, the area enclosed by the parabola and the line, as shaded in the diagram, is equal to \(\dfrac {(p - r)^3 }{ 6}\), where \(p>r\).

    Points R and P are both in the first quadrant with P to the right and below R. An upward opening parabola has its vertex in the fourth quadrant and intersects the positive x axis at two points. The enclosed shaded region is bounded below by the parabola and bounded above by the line through R and P.

    1. Lightbulb What is the area enclosed by the parabola with equation \(y = x^2 + 3x - 12\) and the line with equation \(y = 2x\)?

    2. Full Solution For some values of \(m\), the line with equation \(y = mx - 6\) intersects the parabola with equation \(y = -x^2+7x-90\) at distinct points \(V\) and \(W\) whose \(x\)-coordinates are integers. There are two such values of \(m\) for which the area enclosed by the line and the parabola is as small as possible. Determine these two values of \(m\).

    3. Full Solution Suppose that \(g\) and \(h\) are real numbers for which the parabolas \(y=x^2+(g+h)x+9\) and \(y=-x^2+gx+h\) intersect at distinct points \(T\) and \(U\), as shown.

      Two parabolas, one downward opening and one upward opening and both with vertices in the first quadrant, intersect at points T and U. The region enclosed by the parabolas is shaded and lies entirely in the first quadrant.

      Determine all possible values of \(h\) so that the area enclosed by the parabolas is \(\frac{3087}{8}\).

  4. A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, \(5, 10, 20\) is a geometric sequence with three terms and a common ratio of \(2\).

    1. Lightbulb What is the sum of the first three terms of a geometric sequence with a common ratio of \(\frac{3}{5}\) and whose second term is \(45\)?

    2. Full Solution Determine all pairs of positive integers \((x,y)\) so that \(x, 12, y\) is a geometric sequence and \(x+y=25\).

    3. Full Solution Determine all quadruples of integers \((a, b, c, d)\) so that \(a, b, c, d\) is a geometric sequence and \(a+b+c+d=65\).

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