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)

©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
   • worth 2 or 3 marks each
   • full marks are given for a correct answer which is placed in the box
   • part marks are awarded if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate location in the answer booklet
   • marks awarded for completeness, clarity, and style of presentation
   • a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

• Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages.
• Express answers as simplified exact numbers except where otherwise indicated. For example, \(\pi + 1\) and \(1 - \sqrt{2}\) are simplified exact numbers.

Questions

1. A cyclic quadrilateral is a quadrilateral whose four vertices lie on some circle. In a cyclic quadrilateral, opposite angles add to \(180^{\circ}\). In the diagram, \(ABCD\) is a cyclic quadrilateral. Therefore, \(\angle ABC+\angle ADC=180^{\circ}=\angle BAD+\angle BCD\).

   

   1. In Figure A below, \(ABCD\) is a cyclic quadrilateral.

      

      If \(\angle BAD=88^{\circ}\), what is the value of \(u\)?

   2. In Figure B, \(PQRS\) and \(STQR\) are cyclic quadrilaterals.

      

      If \(\angle STQ=58^{\circ}\), what is the value of \(x\) and what is the value of \(y\)?

   3. In Figure C, \(JKLM\) is a cyclic quadrilateral with \(JK=KL\) and \(JL=LM\).

      

      If \(\angle KJL=35^{\circ}\), what is the value of \(w\)?

   4. In Figure D, \(DEFG\) is a cyclic quadrilateral. \(FG\) is extended to \(H\), as shown.

      

      If \(\angle DEF=z^{\circ}\), determine the measure of \(\angle DGH\) 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 \(n^{th}\) odd integer. The first four rows are given below.

   • Row 1: 1
   • Row 2: 1, 2, 3
   • Row 3: 1, 2, 3, 4, 5
   • Row 4: 1, 2, 3, 4, 5, 6, 7

   In the table, the \(9^{th}\) 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 \(n^2\) integers have been written.

   1. What is the 25\(^{th}\) integer written in the table and in which row does the 25\(^{th}\) integer appear?

   2. What is the 100\(^{th}\) integer written in the table?

   3. What is the 2017\(^{th}\) integer written in the table?

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

3. 1. The line \(y=-15\) intersects the parabola with equation \(y=-x^2+2x\) at two points. What are the coordinates of these two points of intersection?

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

   3. A line intersects the parabola with equation \(y=-x^2+kx\) at \(x=p\) and at \(x=q\) with \(p \ne q\). Determine the \(y\)-intercept of this line.

   4. For all \(k\neq 0\), the curve \(x=\dfrac{1}{k^3}y^2+\dfrac{1}{k}y\) intersects the parabola with equation \(y=-x^2+kx\) 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.

4. A positive integer is called an \(n\)-digit zigzag number if

   • \(3 \leq n \leq 9\),

   • the number’s digits are exactly \(1,2,\ldots,n\) (without repetition), and

   • for each group of three adjacent digits, either the middle digit is greater than each of the other two digits or the middle digit is less than each of the other two digits.

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

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

   2. Let \(G(n,k)\) 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. Determine 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.

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• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
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• Find your school's contest results