2022 Hypatia Contest
(Grade 11)

Tuesday, April 12, 2022
(in North America and South America)

Wednesday, April 13, 2022
(outside of North American and South America)

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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:

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
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, $π + 1$ and $1 - \sqrt{2}$ are simplified exact numbers.

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:

Please read the instructions for the contest.
Write all answers in the answer booklet provided.
For questions marked , place your answer in the appropriate box in the answer booklet and show your work.
For questions marked , 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.
Diagrams are not drawn to scale. They are intended as aids only.
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

A regular hexagon is a polygon that has six sides with equal length and six interior angles with equal measure. In Figure 1, regular hexagon $A B C D E F$ has side length $2 x$ and its vertices lie on the circle with centre $O$ . The diagonals $A D$ , $B E$ and $C F$ divide $A B C D E F$ into six congruent equilateral triangles.
1. In terms of $x$ , what is the radius of the circle?
2. The midpoint of side $A B$ is labelled $M$ , as shown in Figure 2. In terms of $x$ , what is the length of $O M$ ?
3. In terms of $x$ , what is the area of hexagon $A B C D E F$ ?
4. The region that lies inside the circle and outside hexagon $A B C D E F$ is shaded, as shown in Figure 3. The area of this shaded region is 123. Rounded to the nearest tenth, determine the value of $x$ .
With 1 kg of muffin batter, exactly $24$ mini muffins and $2$ large muffins can be made. With 2 kg of muffin batter, exactly $36$ mini muffins and $6$ large muffins can be made.
1. With 2 kg of muffin batter, exactly $48$ mini muffins and $n$ large muffins can also be made. What is the value of $n$ ?
2. With $x$ kg of muffin batter, exactly 84 mini muffins and 10 large muffins can be made. What is the value of $x$ ?
3. Determine how many mini muffins can be made using the same amount of batter that is needed to make 7 large muffins.
A sequence is created in such a way that
- a real number is chosen as the first number in the sequence, and
- each of the following numbers in the sequence is generated by applying a function to the previous number in the sequence.
For example, if the first number in a sequence is 1 and the following numbers are generated by the function $x^{2} - 5$ , then the first three numbers in the sequence are $1, - 4$ and $11$ since $1^{2} - 5 = - 4$ and $(- 4)^{2} - 5 = 11$ .
1. The first number in a sequence is $3$ and the sequence is generated by the function $x^{2} - 3 x + 1$ . What are the first four numbers in the sequence?
2. The number 7 is the third number in a sequence generated by the function $x^{2} - 4 x + 7$ . What are all possible first numbers in the sequence?
3. The first number in a sequence is $c$ and the sequence is generated by the function $x^{2} - 7 x - 48$ . If all numbers in the sequence are equal to $c$ , determine all possible values of $c$ .
4. A sequence generated by the function $x^{2} - 12 x + 39$ alternates between two different numbers. That is, the sequence is $a, b, a, b, a, b, \dots$ , with $a \neq b$ . Determine all possible values of $a$ .
Every integer $N > 1$ can be written as $N$ = $p_{1}^{a_{1}} p_{2}^{a_{2}} p_{3}^{a_{3}} \dots p_{k}^{a_{k}}$ , where $k$ is a positive integer, $p_{1} < p_{2} < p_{3} < \dots < p_{k}$ are prime numbers, and $a_{1}, a_{2}, a_{3}, \dots, a_{k}$ are positive integers. For example, $1400 = 2^{3} 5^{2} 7^{1}$ .

The number of positive divisors of $N$ is denoted by $f (N)$ . It is known that $f (N) = (1 + a_{1}) (1 + a_{2}) (1 + a_{3}) \dots (1 + a_{k})$
1. How many positive divisors does 240 have? That is, what is the value of $f (240)$ ?
2. Define an integer $N > 1$ to be refactorable if it is divisible by $f (N)$ . For example, both $6$ and $8$ have $4$ positive divisors, so $8$ is refactorable and 6 is not refactorable. This is because $8$ is divisible by $4$ , but 6 is not divisible by $4$ . Determine all refactorable numbers $N$ with $f (N) = 6$ .
3. Determine the smallest refactorable number $N$ with $f (N) = 256$ .
4. Show that for every integer $m > 1$ , there exists a refactorable number $N$ such that $f (N) = m$ .

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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