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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)

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©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 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. 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. LightbulbWhich bucket contains 16 red discs?

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

    3. Full solutionThere 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.

    A large square has sides that are 60 centimeters in length. It is divided into one white square and twelve shaded squares. The white square is in the center and the twelve shaded squares form a boarder around the white square. Two shaded squares fit along one edge of the white square and four shaded squares fit along one edge of the 60 by 60 square.

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

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

    2. LightbulbWhen 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. Full solutionA 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 large square has sides that are 60 centimeters in length. It is divided into one white square and forty-eight shaded squares. The white square is in the center, 20 shaded squares surround the white square, and 28 shaded squares surround that. Four shaded squares fit along one edge of the white square and eight shaded squares fit along one edge of the 60 by 60 square.

      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.

    1. LightbulbIn the diagram, \(\triangle ABC\) is equilateral with side length 6 and \(D\) is the midpoint of \(BC\).

      AD meets BC at a right angle. The length of AD is h.

      Determine the exact value of \(h\), the height of \(\triangle ABC\).

    2. LightbulbIn the diagram, a circle with centre \(O\) has radius 6. Regular hexagon \(EFGHIJ\) has sides of length 6 and vertices on the circle.

      Hexagon EFGHIJ is inside the circle. Six line segments connect O to each vertex, dividing the interior of the triangle into six triangular regions. The region inside the circle, but outside the hexagon, is shaded.

      Determine the exact area of the shaded region.

    3. Full solutionA circle has centre \(O\) and radius \(r\). A second circle has centre \(P\) and diameter \(MN\).

      Points M and N lie on the circumference of the circle with centre O. Segments OM, ON and MN are drawn. Point P lies on MN. The circe with centre P, that does through M and N is smaller and overlaps the circle with centre O. The region inside the smaller circle that is outside the larger circle is shaded.

      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 \(n \geq 2\) is of the form \(p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}\) where \(p_1,p_2,\ldots,p_k\) are different prime numbers and \(a_1,a_2,\ldots,a_k\) are positive integers.

    Given an input of an integer \(n\geq 2\), the Barbeau Process outputs the number equal to \(n\left (\dfrac{a_1}{p_1}+\dfrac{a_2}{p_2}+\dfrac{a_3}{p_3}+\cdots+\dfrac{a_k}{p_k} \right)\).

    For example, given an input of 45, the Barbeau Process outputs \(45\left (\dfrac{2}{3}+\dfrac{1}{5}\right )=30+9=39\), since the prime factorization of 45 is \(3^25^1\).

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

    2. Full solutionDetermine all pairs \((p,q)\) of different prime numbers such that the Barbeau Process with input \(p^2q\) outputs 135.

    3. Full solutionDetermine all triples \((a,b,c)\) of positive integers such that the Barbeau Process with input \(2^a3^b5^c\) outputs \(4\times 2^a3^b5^c\).

    4. Full solutionDetermine all integer values of \(n\) with \(2\leq n<10^{10}\) 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

For teachers...

Visit our website cemc.uwaterloo.ca to