2017 Canadian Intermediate
Mathematics Contest

Wednesday, November 22, 2017
(in North America and South America)

Thursday, November 23, 2017
(outside of North American and South America)

©2017 University of Waterloo

Instructions

Time: 2 hours

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.

Do not open this booklet until instructed to do so.
There are two parts to this paper. The questions in each part are arranged roughly in order of increasing difficulty. The early problems in Part B are likely easier than the later problems in Part A.

PART A

1. This part consists of six questions, each worth 5 marks.
2. Enter the answer in the appropriate box in the answer booklet. For these questions, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet.

PART B

1. This part consists of three questions, each worth 10 marks.
2. Finished solutions must be written in the appropriate location in the answer booklet. Rough work should be done separately. If you require extra pages for your finished solutions, they will be supplied by your supervising teacher. Insert these pages into your answer booklet. Write your name, school name, and question number on any inserted pages.
3. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

PART A

For each question in Part A, full marks will be given for a correct answer which is placed in the box. Part marks will be awarded only if relevant work is shown in the space provided in the answer booklet.

1. In the diagram, $BD$ and $AE$ intersect at $C$ and $AB=BC$. Also, $\mathrm{\angle }ABC={40}^{\circ }$ and $\mathrm{\angle }DCE=x^{\circ }$.

   

   What is the value of $x$?

2. There are 12 different four-digit positive integers that can be made by arranging the digits $1,2,7,7$. These integers are listed from smallest to largest. What is the sum of the 6th and 7th integers in the list?

3. In the $3\times 3$ grid shown, each of the three symbols has a different value. The sum of the values of the symbols in each row is given to the right of that row, and the sum of the values of the symbols in each column is given below that column. What is the value of $x$?

   
   Hide/Reveal Description of Grid
   • The first row in order has heart, triangle, square with the number x to the right of the row
   • The second row in order has square, triangle, square with the number 20 to the right of the row
   • The third row in order has heart, triangle, triangle with the number 15 to the right of the row
   • The number 22 appears at the bottom of the first column.
   • The number 12 appears at the bottom of the second column.
   • The number 20 appears at the bottom of the third column.

   ---

4. Yumi has a flat circular chocolate chip cookie with radius 3 cm. On the top of the cookie, there are $k$ circular chocolate chips, each with radius 0.3 cm. No two chocolate chips overlap and no chocolate chip hangs over the edge of the cookie. For what value of $k$ is exactly $\frac{1}{4}$ of the area of the top of the cookie covered with chocolate chips?

5. Positive integers $a$, $b$ and $c$ satisfy the equation ${\displaystyle \frac{31}{72}}={\displaystyle \frac{a}{8}}+{\displaystyle \frac{b}{9}}-c$.
   What is the smallest possible value of $b$?

6. In the diagram, six squares form a $2\times 3$ grid. The middle square in the top row is marked with an R. Each of the five remaining squares is to be marked with an R, S or T. In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter? $$\begin{array}{|ccc|}\hline \phantom{\text{R}}& \text{R}& \phantom{\text{R}}\\ & & \\ \hline\end{array}$$

PART B

For each question in Part B, your solution must be well-organized and contain words of explanation or justification. Marks are awarded for completeness, clarity, and style of presentation. A correct solution, poorly presented, will not earn full marks.

1. The first three figures in a pattern are shown below.

   
   Figure 1       Figure 2         Figure 3

   Figure 1 is formed by three identical squares of side length 1 cm arranged in two rows. The perimeter of Figure 1 is 8 cm. Given a figure in the pattern, the next figure is obtained by adding a square of side length 1 cm on the right-hand end of each of the two rows.

   1. How many squares of side length 1 cm are used to form Figure 8?

   2. Determine the perimeter of Figure 12.

   3. Determine the positive integer $C$ for which the perimeter of Figure $C$ is $38$ cm.

   4. Determine the positive integer $D$ for which the ratio of the perimeter of Figure 29 to the perimeter of Figure $D$ is equal to $\frac{4}{11}$.

2. The symbol $n!$ represents the product of the positive integers from $1$ to $n$. That is, $n!=n\times (n-1)\times (n-2)\times \cdots \times 3\times 2\times 1$. (The symbol $n!$ is read “$n$ factorial”.) For example, the value of $4!$ is 24 because $4\times 3\times 2\times 1=24$.

   1. Determine the value of ${\displaystyle \frac{7!}{5!}}$ .

   2. Determine the positive integer $n$ for which $98!\times 9900=n!$ .

   3. Determine the positive integer $m$ for which ${\displaystyle \frac{(m+2)!}{m!}}=40200$.

   4. Suppose that $q$ is a positive integer and that $r$ is the number for which$(q+2)!-(q+1)!=(q!)\times r$. Show that, for every positive integer $q$, the number $r$ is an integer which is a perfect square.

3. We call a positive integer balanced if

   • it has six digits,

   • each of its six digits is non-zero, and

   • the product of its first three digits is equal to the product of its last three digits.
     

For example, $241181$ is balanced since no digit equals zero and $2\times 4\times 1=1\times 8\times 1$.

1. Determine, with justification, all balanced positive integers of the form $3b8d5f$.

2. Determine, with justification, a three-digit positive integer of the form $4bc$ for which there are exactly three balanced positive integers of the form $4bcdef$.

3. For each of $k=4,5,6,7,8,9,10$, either

   • determine, with justification, an integer $abc$ for which there are exactly $k$ balanced positive integers of the form $abcdef$, or

   • justify why there does not exist an integer $abc$ for which there are exactly $k$ balanced positive integers of the form $abcdef$.