2023 Canadian Senior
Mathematics Contest

Wednesday, November 15, 2023
(in North America and South America)

Thursday, November 16, 2023
(outside of North American and South America)

©2023 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. Two prime numbers $p$ and $q$ satisfy the equation $p+q=31$. What is $pq$?

2. The integer $203$ has an odd ones digit, an even tens digit, and an even hundreds digit. How many integers between $100$ and $999$ have an odd ones digit, an even tens digit, and an even hundreds digit?

3. The distance from point $P(x,y)$ to the origin $O(0,0)$ is $17$.
   The distance from point $P(x,y)$ to $A(16,0)$ is also $17$.
   What are the two possible pairs of coordinates $(x,y)$ for $P$?

4. A store sells shirts, water bottles, and chocolate bars. Shirts cost $\mathrm{\$}10$ each, water bottles cost $\mathrm{\$}5$ each, and chocolate bars cost $\mathrm{\$}1$ each. On one particular day, the store sold $x$ shirts, $y$ water bottles, and $z$ chocolate bars. The total revenue from these sales was $\mathrm{\$}120$. If $x$, $y$ and $z$ are integers with $x>0$, $y>0$ and $z>0$, how many possibilities are there for the ordered triple $(x,y,z)$?

5. What are all pairs of integers $(r,p)$ for which $r^2-r(p+6)+p^2+5p+6=0$?

6. Cube $ABCDEFGH$ has edge length $6$, and has $P$ on $BG$ so that $BP=PG$. What is the volume of the three-dimensional region that lies inside both square-based pyramid $EFGHP$ and square-based pyramid $ABCDG$?

   

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. In the diagram, $ABCD$ is a trapezoid with $\mathrm{\angle }DAB=\mathrm{\angle }ADC=90\mathrm{°}$. Also, $AB=7$, $DC=17$, and $AD=10$. Point $P$ is on $AD$ and point $Q$ is on $BC$ so that $PQ$ is parallel to $AB$ and to $DC$. Point $F$ is on $DC$ so that $BF$ is perpendicular to $DC$. $BF$ intersects $PQ$ at $T$.

   

   1. Determine the area of trapezoid $ABCD$.

   2. Determine the measure of $\mathrm{\angle }BQP$.

   3. If $PQ=x$, determine the length of $AP$ in terms of $x$.

   4. Determine the length of $PQ$ for which the areas of trapezoids $ABQP$ and $PQCD$ are equal.

2. The rectangular region $A$ is in the first quadrant. The bottom side and top side of $A$ are formed by the lines with equations $y=0.5$ and $y=99.5$, respectively. The left side and right side of $A$ are formed by the lines with equations $x=0.5$ and $x=99.5$, respectively.

   1. Determine the number of lattice points that are on the line with equation $y=2x+5$ and are inside the region $A$. (A point with coordinates $(r,s)$ is called a lattice point if $r$ and $s$ are both integers.)

   2. For some integer $b$, the number of lattice points on the line with equation $y=\frac{5}{3}x+b$ and inside the region $A$ is at least $15$. Determine the greatest possible value of $b$.

   3. For some real numbers $m$, there are no lattice points that lie on the line with equation $y=mx+1$ and inside the region $A$. Determine the greatest possible real number $n$ that has the property that, for all real numbers $m$ with $\frac{2}{7}<m<n$, there are no lattice points on the line with equation $y=mx+1$ and inside the region $A$.

3. In this problem, you may choose to consider angles in either degrees or in radians.

   1. Determine an angle $x$ for which $\sin \left({\displaystyle \frac{x}{5}}\right)=\sin \left({\displaystyle \frac{x}{9}}\right)=1$.

   2. There are sequences of $100$ distinct positive integers $n_1$, $n_2$, …, $n_{100}$ with the property that, for all integers $i$ and $j$ with $1\le i<j\le 100$ and for all angles $x$, we have $\sin \left({\displaystyle \frac{x}{n_i}}\right)+\sin \left({\displaystyle \frac{x}{n_j}}\right)\ne 2$. Determine such a sequence $n_1$, $n_2$, $\dots$, $n_{100}$ and prove that it has this property.

   3. Suppose that $m_1,m_2,\dots ,m_{100}$ is a list of $100$ distinct positive integers with the property that, for each integer $i=1,2,\dots ,99$, there is an angle $x_i$ with $\sin \left({\displaystyle \frac{x_i}{m_i}}\right)+\sin \left({\displaystyle \frac{x_i}{m_{i+1}}}\right)=2$. Prove that, for every such sequence $m_1,m_2,\dots ,m_{100}$ with $m_1=6$, there exists an angle $t$ for which $$\sin \left({\displaystyle \frac{t}{m_1}}\right)+\sin \left({\displaystyle \frac{t}{m_2}}\right)+\cdots +\sin \left({\displaystyle \frac{t}{m_{100}}}\right)=100$$ (The sum on the left side consists of 100 terms.)