2016 Galois Contest
(Grade 10)

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

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

©2016 University of Waterloo

Instructions

Time: $75$ minutes

Number of Questions: 10

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. Liza has a row of buckets. The first bucket contains 17 green discs and 7 red discs. Each bucket after the first contains 1 more green disc and 3 more red discs than the previous bucket.

   1. Which bucket contains 16 red discs?

   2. In which bucket is the number of red discs equal to the number of green discs?

   3. There is a bucket in which the number of red discs is twice the number of green discs. In total, how many discs are in this bucket?

2. Judy has square plates, each with side length 60 cm. A plate is Shanks-Decorated if identical shaded squares are drawn along the outer edges of the plate, as shown.

   

   The diagram shows an example of a plate that is Shanks-Decorated with 12 shaded squares.

   1. Judy’s first plate is Shanks-Decorated with 36 shaded squares. What is the side length of each shaded square?

   2. When a second plate is Shanks-Decorated, an area of 1600 cm${}^2$ is left unshaded in the centre of the plate. What is the side length of each shaded square?

   3. A plate is Double-Shanks-Decorated if two layers of identical shaded squares are drawn along the outer edges of the plate, as shown. The diagram shows an example of a plate that is Double-Shanks-Decorated with 48 shaded squares.

      

      A new plate is Double-Shanks-Decorated and an area of 2500 cm${}^2$ is left unshaded in the centre of the plate. Determine the number of shaded squares.

3. 1. In the diagram, $\mathrm{△}ABC$ is equilateral with side length 6 and $D$ is the midpoint of $BC$.

      

      Determine the exact value of $h$, the height of $\mathrm{△}ABC$.

   2. In the diagram, a circle with centre $O$ has radius 6. Regular hexagon $EFGHIJ$ has sides of length 6 and vertices on the circle.

      

      Determine the exact area of the shaded region.

   3. A circle has centre $O$ and radius $r$. A second circle has centre $P$ and diameter $MN$.

      

      The circles intersect at $M$ and $N$. If $MN=r$, determine the exact area of the shaded region, in terms of $r$.

4. The prime factorization of 45 is $3^2{5}^1$. In general, the prime factorization of an integer $n\ge 2$ is of the form $p_1^{a_1}{p}_2^{a_2}{p}_3^{a_3}\cdots p_k^{a_k}$ where $p_1,p_2,\dots ,p_k$ are different prime numbers and $a_1,a_2,\dots ,a_k$ are positive integers.

   Given an input of an integer $n\ge 2$, the Barbeau Process outputs the number equal to $n({\displaystyle \frac{a_1}{p_1}}+{\displaystyle \frac{a_2}{p_2}}+{\displaystyle \frac{a_3}{p_3}}+\cdots +{\displaystyle \frac{a_k}{p_k}})$.

   For example, given an input of 45, the Barbeau Process outputs $45({\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{5}})=30+9=39$, since the prime factorization of 45 is $3^2{5}^1$.

   1. Given an input of 126, what number does the Barbeau Process output?

   2. Determine all pairs $(p,q)$ of different prime numbers such that the Barbeau Process with input $p^{2}q$ outputs 135.

   3. Determine all triples $(a,b,c)$ of positive integers such that the Barbeau Process with input $2^a{3}^b{5}^c$ outputs $4\times 2^a{3}^b{5}^c$.

   4. Determine all integer values of $n$ with $2\le n<{10}^{10}$ such that the Barbeau Process with input $n$ outputs $3n$.

Further Information

For students...

Thank you for writing the Galois 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

• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
• Information about careers in and applications of mathematics and computer science

For teachers...

Visit our website cemc.uwaterloo.ca to

• Obtain information about future contests
• Look at our free online courseware for high school students
• Learn about our face-to-face workshops and our web resources
• Subscribe to our free Problem of the Week
• Investigate our online Master of Mathematics for Teachers
• Find your school's contest results