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2026 Hypatia Contest
(Grade 11)

Wednesday, April 1, 2026
(in North America and South America)

Thursday, April 2, 2026
(outside of North American and South America)

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©2026 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 Short answer
  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 Short answer, 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. If points are collinear, then they lie on the same straight line.

    1. Short answer If the points \(A(1,2)\), \(B(2,5)\) and \(C(3,c)\) are collinear, what is the value of \(c\)?

    2. Short answer Three distinct points \(D(0,7)\), \(E\) and \(F(14,0)\) are collinear. If the coordinates of point \(E\) are positive integers, how many such points are possible?

    3. Full solution Determine the value of \(n\) so that the points \(P(15,12)\), \(Q(6,n-4)\) and \(R(18,n)\) are collinear.

  2. A peak number is a 5-digit positive integer, \(ABCBA\), with digits \(0< A < B < C<10\). For example, \(27\,872\) is a peak number with \(A=2\), \(B=7\) and \(C=8\), but \(52\,625\) and \(46\,954\) are not peak numbers.

    1. Short answer What is the positive difference between the largest and smallest peak numbers?

    2. Short answer How many peak numbers are greater than \(36\,245\) and less than 45 932?

    3. Full solution Determine all peak numbers that are a multiple of \(15\).

  3. In the diagram, the circles with centres \(P\) and \(Q\) are externally tangent to one another at \(V\), the circles with centres \(P\) and \(R\) are internally tangent to one another at \(W\), and \(ST\) is tangent to the circle with centre \(Q\) at \(U\).

    Fact 1: The line through the centres of two circles that are tangent to one another passes through their point of tangency. For example, in the diagram, the line through \(P\) and \(Q\) passes through \(V\), and the line through \(P\) and \(R\) passes through \(W\).

    Fact 2: A line tangent to a circle is perpendicular to the radius drawn to the point of tangency. For example, in the diagram, \(ST\) is perpendicular to \(QU\).

    1. Short answer In Figure 1 below, a small circle with centre \(A\) and radius \(10\) is drawn inside a large circle with centre \(B\) and radius \(36\). The two circles are internally tangent to one another at \(C\). Diameter \(CD\) of the large circle passes through \(A\) and intersects the small circle at \(E\). What is the length of \(DE\)?

    2. Full solution In Figure 2 below, a small circle with centre \(G\) and radius \(12\) is drawn inside a large circle with centre \(F\) and radius \(49\). The two circles are internally tangent to one another at \(K\). Diameter \(HI\) of the large circle is tangent to the small circle at \(J\). Determine the length of \(FJ\).

    3. Full solution In Figure 3 below, a small circle with centre \(X\) and radius \(5\) is drawn inside a large circle with centre \(Z\) and radius \(18\). These two circles are internally tangent to one another at \(L\). Diameter \(UV\) of the large circle is tangent to the small circle at \(M\). A medium circle with radius \(r\) is externally tangent to the small circle at \(N\), internally tangent to the large circle at \(V\), and has centre \(Y\) on \(UV\). Determine the value of \(r\).

      Figure 1

      Figure 2

      Figure 3

  4. A positive integer \(n\) is a sum of two positive squares if there exist integers \(a>0\) and \(b>0\) for which \(n = a^2 + b^2\). For example, \(29\) is a sum of two positive squares since \(29 = 2^2 + 5^2\). However, each of \(11\) and \(16\) is not a sum of two positive squares. A Sorensen number, \(S\), is a positive integer that can be written as a sum of two positive squares in the following three distinct ways: \[\begin{align*} S & =m^2+x^2=(m+1)^2+y^2=(m+2)^2+z^2 \tag{$\ast$}\end{align*}\] for some positive integers \(m\), \(x\), \(y\), and \(z\). In the questions that follow, \(x\), \(y\) and \(z\) are positive integers satisfying \((\ast)\).

    1. Short answer If \(y^2-z^2=x^2-y^2+k\), what is the value of the constant \(k\)?

    2. Full solution The expression \(y^2-z^2\) can be written as a sum of \(p\) consecutive odd integers, the largest of which is \(2y-1\). Similarly, \(x^2-y^2\) can be written as a sum of \(q\) consecutive odd integers, the largest of which is \(2x-1\). Determine an expression for \(y\) written in terms of \(p\) and \(q\).

    3. Full solution Determine two different Sorensen numbers.

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