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2017 Hypatia Contest
(Grade 11)

Wednesday, April 12, 2017
(in North America and South America)

Thursday, April 13, 2017
(outside of North American and South America)

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©2017 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=x3−x, you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

Questions

  1. A cyclic quadrilateral is a quadrilateral whose four vertices lie on some circle. In a cyclic quadrilateral, opposite angles add to 180∘. In the diagram, ABCD is a cyclic quadrilateral. Therefore, ∠ABC+∠ADC=180∘=∠BAD+∠BCD.

    1. LightbulbIn Figure A below, ABCD is a cyclic quadrilateral.

      Angle BCD is 2 u degrees.

      If ∠BAD=88∘, what is the value of u?

    2. LightbulbIn Figure B, PQRS and STQR are cyclic quadrilaterals.

      Starting at P and moving around the circle, the points are P, S, R, Q, and T, in order. Angle SRQ measures x degrees. Angle SPQ measures y degrees.

      If ∠STQ=58∘, what is the value of x and what is the value of y?

    3. LightbulbIn Figure C, JKLM is a cyclic quadrilateral with JK=KL and JL=LM.

      Angle JLM measures w degrees.

      If ∠KJL=35∘, what is the value of w?

    4. Full solutionIn Figure D, DEFG is a cyclic quadrilateral. FG is extended to H, as shown.

      H is outside of the circle on which the vertices D, E, F, and G lie.

      If ∠DEF=z∘, determine the measure of ∠DGH in terms of z.

  2. A list of integers is written in a table, row after row from left to right. Row 1 has the integer 1. Row 2 has the integers 1,2 and 3. Row n has the consecutive integers beginning at 1 and ending at the nth odd integer. The first four rows are given below.

    In the table, the 9th integer to be written is 5, and it appears at the end of Row 3. In general, after having completed n rows, a total of n2 integers have been written.

    1. LightbulbWhat is the 25th integer written in the table and in which row does the 25th integer appear?

    2. LightbulbWhat is the 100th integer written in the table?

    3. Full solutionWhat is the 2017th integer written in the table?

    4. Full solutionIn how many of the first 200 rows does the integer 96 appear?

    1. LightbulbThe line y=−15 intersects the parabola with equation y=−x2+2x at two points. What are the coordinates of these two points of intersection?

    2. Full solution A line intersects the parabola with equation y=−x2−3x at x=4 and at x=a. This line intersects the y-axis at (0,8). Determine the value of a.

    3. Full solutionA line intersects the parabola with equation y=−x2+kx at x=p and at x=q with p≠q. Determine the y-intercept of this line.

    4. Full solutionFor all k≠0, the curve x=1k3y2+1ky intersects the parabola with equation y=−x2+kx at (0,0) and at a second point T whose coordinates depend on k. All such points T lie on a parabola. Determine the equation of this parabola.

  3. A positive integer is called an n-digit zigzag number if

    For example, 52314 is a 5-digit zigzag number but 52143 is not.

    1. LightbulbWhat is the largest 9-digit zigzag number?

    2. Full solutionLet G(n,k) be the number of n-digit zigzag numbers with first digit k and second digit greater than k. Let L(n,k) be the number of n-digit zigzag numbers with first digit k and second digit less than k.

      1. Show that G(6,3)=L(5,3)+L(5,4)+L(5,5).

      2. Show that G(6,1)+G(6,2)+G(6,3)+G(6,4)+G(6,5)+G(6,6) equals L(6,1)+L(6,2)+L(6,3)+L(6,4)+L(6,5)+L(6,6).

    3. Full solutionDetermine the number of 8-digit zigzag 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