2015 Canadian Senior
Mathematics Contest

Wednesday, November 25, 2015
(in North America and South America)

Thursday, November 26, 2015
(outside of North American and South America)

©2015 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. If ${\displaystyle \frac{8}{24}}={\displaystyle \frac{4}{x+3}}$, what is the value of $x$?

2. Let $A$, $B$ and $C$ be non-zero digits, so that $BC$ is a two-digit positive integer and $ABC$ is a three-digit positive integer made up of the digits $A$, $B$ and $C$. Suppose that $$\begin{array}{cccc}& & B& C\\ & A& B& C\\ +& A& B& C\\ & 8& 7& 6\end{array}$$ What is the value of $A+B+C$?

3. A 5 m $\times$ 5 m flat square roof receives 6 mm of rainfall. All of this water (and no other water) drains into an empty rain barrel. The rain barrel is in the shape of a cylinder with a diameter of 0.5 m and a height of 1 m. Rounded to the nearest tenth of a percent, what percentage of the barrel will be full of water?

4. Determine all values of $x$ for which ${(2\cdot 4^{x^2-3x})}^2=2^{x-1}$.

5. In a psychology experiment, an image of a cat or an image of a dog is flashed briefly onto a screen and then Anna is asked to guess whether the image showed a cat or a dog. This process is repeated a large number of times with an equal number of images of cats and images of dogs shown. If Anna is correct 95% of the time when she guesses “dog" and 90% of the time when she guesses “cat", determine the ratio of the number of times she guessed “dog" to the number of times she guessed “cat".

6. Suppose that $X$ and $Y$ are angles with $\tan X={\displaystyle \frac{1}{m}}$ and $\tan Y={\displaystyle \frac{a}{n}}$ for some positive integers $a$, $m$ and $n$. Determine the number of positive integers $a\le 50$ for which there are exactly 6 pairs of positive integers $(m,n)$ with $X+Y={45}^{\circ }$.
   (Note: The formula  $\tan (X+Y)={\displaystyle \frac{\tan X+\tan Y}{1-\tan X\tan Y}}$  may be useful.)
   

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 line $y=2x+4$ intersects the $y$-axis at $R$, as shown. A second line, parallel to the $y$-axis, is drawn through $P(p,0)$, with $p>0$. These two lines intersect at $Q$.

   

   • Determine the length of $OR$.
     (Note that $O$ is the origin $(0,0)$.)

   • Determine the coordinates of point $Q$ in terms of $p$.

   • If $p=8$, determine the area of $OPQR$.

   • If the area of $OPQR$ is 77, determine the value of $p$.

2. 1. If $f(x)=\frac{x}{x-1}$ for $x\ne 1$, determine all real numbers $r\ne 1$ for which $f(r)=r$.

   2. If $f(x)=\frac{x}{x-1}$ for $x\ne 1$, show that $f(f(x))=x$ for all real numbers $x\ne 1$.

   3. Suppose that $k$ is a real number. Define $g(x)=\frac{2x}{x+k}$ for $x\ne -k$. Determine all real values of $k$ for which $g(g(x))=x$ for every real number $x$ with $x\ne -k$ and $g(x)\ne -k$.

   4. Suppose that $a$, $b$ and $c$ are non-zero real numbers. Define $h(x)={\displaystyle \frac{ax+b}{bx+c}}$ for $x\ne -{\displaystyle \frac{c}{b}}$. Determine all triples $(a,b,c)$ for which $h(h(x))=x$ for every real number $x$ with $x\ne -{\displaystyle \frac{c}{b}}$ and $h(x)\ne -{\displaystyle \frac{c}{b}}$.
      

3. Given a sequence $a_1,a_2,a_3,\dots$ of positive integers, we define a new sequence $b_1,b_2,b_3,\dots$ by $b_1=a_1$ and, for every positive integer $n\ge 1$, $$b_{n+1}=\{\begin{array}{ll}{b}_n+a_{n+1}& \text{if }{b}_n\le a_{n+1}\\ b_n-a_{n+1}& \text{if }{b}_n>a_{n+1}\end{array}$$ For example, when $a_1,a_2,a_3,\cdots$ is the sequence $1,2,1,2,1,2,\dots$ we have

   $n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$\cdots$
   $a_n$	$1$	$2$	$1$	$2$	$1$	$2$	$1$	$2$	$1$	$2$	$\cdots$
   $b_n$	$1$	$3$	$2$	$4$	$3$	$1$	$2$	$4$	$3$	$1$	$\cdots$

   1. Suppose that $a_n=n^2$ for all $n\ge 1$. Determine the value of $b_{10}$.

   2. Suppose that $a_n=n$ for all $n\ge 1$. Determine all positive integers $n$ with $n<2015$ for which $b_n=1$.

   3. A sequence $x_1,x_2,x_3,\dots$ is called eventually periodic if there exist positive integers $r$ and $p$ for which $x_{n+p}=x_n$ for all $n\ge r$.
      Suppose that $a_1,a_2,a_3,\dots$ is eventually periodic. Prove that $b_1,b_2,b_3,\dots$ is eventually periodic.