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2020 Galois Contest
(Grade 10)

Wednesday, April 15, 2020
(in North America and South America)

Thursday, April 16, 2020
(outside of North American and South America)

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©2020 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 \(y=x^{3} -x\), you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

Questions

  1. The letters \(A\) and \(B\) are used to create a pattern consisting of a number of rows. The pattern starts with a single \(A\). The rows alternate between \(A\)’s and \(B\)’s, and the number of letters in each row is twice the number of letters in the previous row. The first 4 rows of the pattern are shown.

    Row 1 \(A\)
    Row 2 \(BB\)
    Row 3 \(A\)\(A\)\(A\)\(A\)
    Row 4 \(BBBBBBBB\)

    1. LightbulbIf the pattern consists of 6 rows, how many letters are in the \(6^{th}\) row of the pattern?

    2. LightbulbIf the pattern consists of 6 rows, what is the total number of letters in the pattern?

    3. Full solutionIf the total number of letters in the pattern is \(63\), determine the number of \(A\)’s in the pattern and the number of \(B\)’s in the pattern.

    4. Full solutionIf the total number of letters in the pattern is 4095, determine the difference between the number of \(A\)’s and the number of \(B\)’s in the pattern.

  2. For a rectangular prism with length \(\ell\), width \(w\), and height \(h\) as shown, the surface area is given by the formula \(A = 2\ell w + 2\ell h + 2wh\) and the volume is given by the formula \(V = \ell wh\).

    1. LightbulbWhat is the surface area of a rectangular prism with length \(2\) cm, width \(5\) cm, and height \(9\) cm?

    2. LightbulbA rectangular prism with height 10 cm has a square base. The volume of the prism is 160 cm\(^3\). What is the side length of the square base?

    3. Full solutionA rectangular prism has a square base with area \(36\) cm\(^2\). The surface area of the prism is \(240\) cm\(^2\). Determine the volume of the prism.

    4. Full solutionA rectangular prism has length \(k\) cm, width \(2k\) cm, and height \(3k\) cm, where \(k > 0\). The volume of the prism is \(x \mbox{ cm}^3\). The surface area of the prism is \(x \mbox{ cm}^2\). Determine the value of \(k\).

  3. Jodi multiplied the numbers \(2\) and \(5\) to get a product of \(10\). She added \(4\) to each of her original numbers to get \(6\) and \(9\). She multiplied these new numbers to get a product of \(54\). Jodi noticed that each of the digits in the new product, \(54\), was \(4\) more than the corresponding digits in the first product, \(10\).

    The pair \((2,5)\) is an example of a RadPair.

    In general, a pair of positive integers \((a, b)\) with \(a \leq b \leq 9\) and for which the product \(ab\) is a two-digit integer is called a RadPair if there exists a positive integer \(d\) such that

    1. Full solutionShow that \((2, 8)\) is a RadPair.

    2. Full solutionShow that \((3,6)\) is not a RadPair.

    3. LightbulbFor which positive integers \(x\) with \(x\le 6\) is \((x, 6)\) a RadPair?

    4. Full solutionDetermine, with justification, the number of RadPairs \((a,b)\) with \(a\le b\).

  4. In an \(n \times n\) grid of unit squares, each point at which two grid lines meet is called a vertex, and so there are \((n+1)^2\) vertices. The top left corner vertex is labeled \(A\) and the bottom right corner vertex is labeled \(B\). A path from \(A\) to \(B\) is a sequence of unit edges that

    The length of such a path is the number of unit edges in the path. For example, in a \(3\times 3\) grid, a path of length 12 between \(A\) and \(B\) is shown.

    A path can be described as a sequence of moves, down (D), up (U), right (R), left (L), between adjacent vertices, beginning at A and ending at B. In the 3 by 3 grid that is shown, the path of length 12 is as follows: R D L D R R U U R D D D.

    1. LightbulbIn a \(2\times 2\) grid, determine the number of paths of any length from \(A\) to \(B\).

    2. Full solutionExplain why there cannot be a path from \(A\) to \(B\) of odd length in a \(10 \times 10\) grid.

    3. Full solutionIn a \(4\times 4\) grid, determine the number of paths of length 10 from \(A\) to \(B\).

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