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)

University of Waterloo Logo

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

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 $B B$

Row 3 $A$ $A$ $A$ $A$

Row 4 $B B B B B B B B$
1. If the pattern consists of 6 rows, how many letters are in the $6^{t h}$ row of the pattern?
2. If the pattern consists of 6 rows, what is the total number of letters in the pattern?
3. If 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. If 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.
For a rectangular prism with length $ℓ$ , width $w$ , and height $h$ as shown, the surface area is given by the formula $A = 2 ℓ w + 2 ℓ h + 2 w h$ and the volume is given by the formula $V = ℓ w h$ .
1. What is the surface area of a rectangular prism with length $2$ cm, width $5$ cm, and height $9$ cm?
2. A 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. A 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. A rectangular prism has length $k$ cm, width $2 k$ cm, and height $3 k$ cm, where $k > 0$ . The volume of the prism is $x {cm}^{3}$ . The surface area of the prism is $x {cm}^{2}$ . Determine the value of $k$ .
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 $a b$ is a two-digit integer is called a RadPair if there exists a positive integer $d$ such that
- the product $(a + d) (b + d)$ is a two-digit integer, and
- the ones (units) digit of the product $(a + d) (b + d)$ equals $d$ plus the ones digit of the product $a b$ , and
- the tens digit of the product $(a + d) (b + d)$ equals $d$ plus the tens digit of the product $a b$ .
1. Show that $(2, 8)$ is a RadPair.
2. Show that $(3, 6)$ is not a RadPair.
3. For which positive integers $x$ with $x \leq 6$ is $(x, 6)$ a RadPair?
4. Determine, with justification, the number of RadPairs $(a, b)$ with $a \leq b$ .
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
- each connect two adjacent vertices, and
- when connected, form a sequence of vertices that begins at $A$ , ends at $B$ , and
- passes through each vertex at most once.
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.
1. In a $2 \times 2$ grid, determine the number of paths of any length from $A$ to $B$ .
2. Explain why there cannot be a path from $A$ to $B$ of odd length in a $10 \times 10$ grid.
3. In a $4 \times 4$ grid, determine the number of paths of length 10 from $A$ to $B$ .

Row 1	$A$
Row 2	$B B$
Row 3	$A$ $A$ $A$ $A$
Row 4	$B B B B B B B B$

Further Information

For students...

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