2016 Canadian Senior
Mathematics Contest

Wednesday, November 23, 2016
(in North America and South America)

Thursday, November 24, 2016
(outside of North American and South America)

©2016 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. What is the value of  ${\displaystyle \frac{1^3+2^3+3^3}{(1+2+3{)}^2}}$ ?

2. Mike bought two books from the Canadian Super Mathematics Company. He paid full price for a $33 book and he received 50% off the full price of a second book. In total, he saved 20% on his purchase. In dollars, how much did he save?

3. How many different lists $a,b,c,d$ of distinct odd positive integers with $a<b<c<d$ have the property that $a+b+c+d=24$?

4. The figure (called a net) has dimensions as shown.

   
   

   
   Hide/Reveal Description of Figure

   There are six rectangles in the net: 2 large, 2 medium and 2 small.

   • The small and medium rectangles are placed end to end, with their longer sides horizontal, in the order small, medium, small medium to form the middle of the net. These rectangles have a total length of 38.
   • One large rectangle is placed above the middle. Its shorter side aligns with a longer side of one of the small rectangles. The vertical distance from the bottom edge of the middle rectangles to the top of the net is 14.
   • The second large rectangle is placed below the middle. Its longer side aligns with a longer side of one of the medium rectangles. The vertical distance from the top edge of the middle rectangles to the bottom of the net is 11.

   ---

   This net can be folded to form a rectangular prism. What is the volume of the prism?    

5. Gary and Deep play a game in which there are no ties. Each player is equally likely to win each game. The first player to win 4 games becomes the champion, and no further games are played. Gary wins the first two games. What is the probability that Deep becomes the champion?

6. A nine-digit positive integer $n$ is called a Moffat number if

   • its digits are $0,1,2,3,4,5,6,7,8$ with no repetition,

   • the sum of each group of 5 digits in a row is divisible by 5, and

   • the sum of each group of 7 digits in a row is divisible by 4.

   For example, $578460213$ is not a Moffat number because, while the sum of each group of 7 digits in a row is divisible by 4, not all of the sums of groups of 5 digits in a row are divisible by 5. Determine the sum of all Moffat numbers.

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. A positive integer that is the same when read forwards or backwards is called a palindrome. A table is created, as shown, in which the first column contains all palindromes between 1000 and 10 000, listed in increasing order. The second column contains the positive difference between each pair of consecutive palindromes. For example, 1001 and 1111 are consecutive palindromes and their positive difference is $1111-1001=110$.

   Palindrome	Difference from last
   1001
   1111	110
   1221	110
   1331	110
   1441	110
   $⋮$	$⋮$

   1. What are the eighth and ninth palindromes in the first column of the table?

   2. The second column of the table contains only two different numbers: 110 and $x$. Determine the value of $x$.

   3. When the table is complete, there are $N$ palindromes in the first column. What is the value of $N$?

   4. Determine the average of all of the $N-1$ numbers in the second column of the table.
      

2. In the diagram, the circle shown has centre $C(6,8)$, has radius 10, and has equation $$(x-6{)}^2+(y-8{)}^2=100$$ The circle passes through the origin $O(0,0)$.

   

   1. Determine the coordinates of the point $P$ where the circle intersects the $x$-axis.

   2. Determine, without justification, the coordinates of the point $Q$ on the circle that has the maximum $y$-coordinate among all points on the circle.

   3. Determine the coordinates of the point $R$ on the circle so that $\mathrm{\angle }PQR={90}^{\circ }$.

   4. Determine the coordinates of the two distinct points $S$ and $T$ on the circle so that $\mathrm{\angle }PQS=\mathrm{\angle }PQT={45}^{\circ }$.
      

3. For each positive integer $n$, define $a_n$ and $b_n$ to be the positive integers such that $$(\sqrt{3}+\sqrt{2}{)}^{2n}=a_n+b_n\sqrt{6}\text{and}(\sqrt{3}-\sqrt{2}{)}^{2n}=a_n-b_n\sqrt{6}$$

   1. Determine the values of $a_2$ and $b_2$.

   2. Prove that $2a_n-1<(\sqrt{3}+\sqrt{2}{)}^{2n}<2a_n$ for all positive integers $n$.

   3. Let $d_n$ be the ones (units) digit of the number $(\sqrt{3}+\sqrt{2}{)}^{2n}$ when it is written in decimal form. Determine, with justification, the value of $$d_1+d_2+d_3+\cdots +d_{1865}+d_{1866}+d_{1867}$$ (The given sum has 1867 terms.)