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2024 Hypatia Contest
(Grade 11)

Thursday, April 4, 2024
(in North America and South America)

Friday, April 5, 2024
(outside of North American and South America)

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©2024 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. At Radford Motors, \(4050\) trucks were sold. Of the trucks sold, \(32\%\) were white, \(24\%\) were grey, and \(44\%\) were black.

    1. Lightbulb How many white trucks were sold?

    2. Lightbulb If \(\dfrac 14\) of the grey trucks sold were electric, how many trucks sold were both grey and electric?

    3. Full solution In addition to the \(4050\) trucks that were sold, there were \(k\) unsold trucks, all of which were black. In total, \(46\%\) of all trucks, sold and unsold, were black. Determine the value of \(k\).

  2. For a positive 3-digit integer \(n\), \(f(n)\) is equal to the sum of \(n\) and the digits of \(n\). For example, \(f(351)=351+3+5+1=360\).

    Note: The decimal representation of the 3-digit number \(abc\) is \(a\cdot10^2+b\cdot10+c\). For example, \(836=8\cdot 10^2+3\cdot 10+6\).

    1. Lightbulb What is the value of \(f(132)\)?

    2. Lightbulb If \(f(n)=175\), what is the value of \(n\)?

    3. Full solution If \(f(n)=204\), determine all possible values of \(n\).

  3. In the diagram, \(ABCD\) is a square with side length \(12\). The midpoint of \(AD\) is \(E\), and \(BE\) intersects \(AC\) at \(F\).

    ABCD is plotted in the first quadrant of the Cartesian plane. A has coordinates (0,0), B has coordinates (12,0), C has coordinates (12,12), and D has coordinates (0,12). E, the midpoint of AD, has coordinates (0,6).

    The circle with diameter \(BE\) passes through \(A\), and intersects \(AC\) at \(G\).

    Note: A circle with centre \((h, k)\) and radius \(r\) has equation \((x - h)^2 + (y - k)^2 = r^2\).

    1. Lightbulb What are the coordinates of \(F\)?

    2. Lightbulb What is the area of \(\triangle AEF\)?

    3. Full solution Determine the area of quadrilateral \(GDEF\).

  4. A Hewitt number is a positive integer that is the sum of the cubes of three consecutive positive integers. The smallest Hewitt number is \(1^3+2^3+3^3=36\).

    1. Lightbulb How many Hewitt numbers between \(10\,000\) and \(100\,000\) are divisible by \(10\)?

    2. Full solution Determine how many of the smallest \(2024\) Hewitt numbers are divisible by \(216\).

    3. Full solution Consider the following statement:

      There are two distinct Hewitt numbers whose sum is equal to \(9\cdot 2^k\) for some positive integer \(k\).

      Show that this statement is true by finding two such Hewitt numbers or prove that it is false by demonstrating that there cannot be two such Hewitt 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