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2016 Hypatia Contest
(Grade 11)

Thursday, April 13, 2016
(in North America and South America)

Friday, April 14, 2016
(outside of North American and South America)

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©2016 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. Raisins are sold by the scoop, cup, jar, basket, or tub in the following proportions:
    5 scoops of raisins fill 1 jar, 3 scoops of raisins fill 1 cup, 5 baskets of raisins fill 2 tubs, and 30 jars of raisins fill 1 tub.

    1. LightbulbHow many tubs of raisins fill 30 baskets?

    2. LightbulbHow many cups of raisins fill 6 jars?

    3. Full solutionDetermine how many cups of raisins fill 1 basket.

  2. If a line segment is drawn from the centre of a circle to the midpoint of a chord, it is perpendicular to that chord. For example, in Figure 1, \(OM\) is perpendicular to chord \(AB\).

    If a line segment is drawn from the centre of a circle and is perpendicular to a chord, it passes through the midpoint of that chord. For example, in Figure 2, \(PR=QR\).

    A circle centred at the point O. The chord AB is drawn horizontally, and its midpoint is denoted M. A vertical line is drawn from O to M, and is shown to be perpendicular to AB.

    A circle centred at the point O. A chord PQ is drawn in the circle, with its midpoint denoted R. A line is drawn from O to R that is shown to be perpendicular to PQ.

    1. LightbulbIn the diagram, a circle with radius 13 has a chord \(AB\) with length 10.

      In the diagram, point O is the centre of the circle, and radius OA is drawn.

      If \(M\) is the midpoint of \(AB\), what is the length of \(OM\)?

    2. LightbulbIn a circle with radius 25, a chord is drawn so that its perpendicular distance from the centre of the circle is 7. What is the length of this chord?

    3. Full solutionIn the diagram, the radius of the circle is 65. Two parallel chords \(ST\) and \(UV\) are drawn so that the perpendicular distance between the chords is 72 (\(MN=72\)).

      M lies on ST and N lies on UV such that MN is perpendicular to both ST nand UV.

      If \(MN\) passes through the centre of the circle \(O\), and \(ST\) has length 112, determine the length of \(UV\).

  3. For a positive integer \(n\), \(f(n)\) is defined as the exponent of the largest power of \(3\) that divides \(n\).
    For example, \(f(126) = 2\) since \(126 = 3^2 \times 14\) so \(3^2\) divides \(126\), but \(3^3\) does not.

    1. LightbulbWhat is the value of \(f(405)\)?

    2. LightbulbWhat is the value of \(f(1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10)\)?

    3. Full solutionLet \(N\) be the positive integer equal to \(\dfrac{100!}{50!20!}\). Determine the value of \(f(N)\).
      (Note: If \(m\) is a positive integer, \(m!\) represents the product of the integers from 1 to \(m\), inclusive. For example, \(5!=1 \times 2 \times 3 \times 4 \times 5=120\).)

    4. Full solutionGiven that \(f(a) = 8\) and \(f(b) = 7\), determine all possible values of \(f(a+b)\).

  4. Erin’s Pizza (EP) and Lino’s Pizza (LP) are located next door to each other. Each day, each of 100 customers buys one whole pizza from one of the restaurants. The price of a pizza at each restaurant is set each day and is always a multiple of 10 cents. If the two restaurants charge the same price, half of the 100 customers will go to each restaurant. For every 10 cents that one restaurant’s price is higher than the other restaurant’s price, it loses one customer to the other restaurant. The cost for each restaurant to make a pizza is $5.00.

    As an example, if EP charges $8.00 per pizza and LP charges $9.00 per pizza, the number of customers and the resulting profit for each restaurant is shown in the table below.

    Restaurant Price per pizza Number of customers Profit
    EP $8.00 \(50+10=60\) \(60\times(\$8.00-\$5.00)=\$180\)
    LP $9.00 \(50-10=40\) \(40\times(\$9.00-\$5.00)=\$160\)

    1. LightbulbOn Monday, EP charges $7.70 for a pizza and LP charges $9.30.

      1. How many customers does LP have?

      2. What is LP’s total profit?

    2. Full solutionEP sets its price first and then LP sets its price. On Tuesday, EP charges $7.20 per pizza. What should LP’s price be in order to maximize its profit?

    3. Full solutionOn Wednesday, EP realizes what LP is doing: LP is maximizing its profit by setting its price after EP’s price is set. If EP continues to set its price first, determine the two prices that EP could charge in order to maximize its profit. State LP’s profit in each case.

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