2021 Canadian Senior
Mathematics Contest

Wednesday, November 17, 2021
(in North America and South America)

Thursday, November 18, 2021
(outside of North American and South America)

©2021 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, a row of 3 squares is made using 10 toothpicks.

   

   In total, how many toothpicks are needed to make a row of 11 squares?

2. The operation $\mathrm{\nabla }$ is defined by $a\mathrm{\nabla }b=(a+1)(b-2)$ for real numbers $a$ and $b$. For example, $4\mathrm{\nabla }5=(4+1)(5-2)=15$. If $5\mathrm{\nabla }x=30$, what is the value of $x$?

3. Consider the points $P(0,0)$, $Q(4,0)$ and $R(1,2)$. The line with equation $y=mx+b$ is parallel to $PR$ and passes through the midpoint of $QR$. What is the value of $b$?

4. In an experiment, 1000 people receive Medicine A and 1000 different people receive Medicine B. The 2000 people are asked whether they have severe side effects, mild side effects, or no side effects. The following information is obtained from the experiment:

   1. The probability that a random person has severe side effects is $\frac{3}{25}$.

   2. The probability that a random person with severe side effects was given Medicine A is $\frac{2}{3}$.

   3. The probability that a random person who was given Medicine A has severe or mild side effects is $\frac{19}{100}$.

   4. The probability that a random person who was given Medicine B has severe or mild side effects is $\frac{3}{20}$.

   What is the probability that a random person with mild side effects was given Medicine B?

5. What are all real numbers $x>0$ for which ${\log }_2(x^2)+2{\log }_{x}8={\displaystyle \frac{392}{{\log }_2(x^3)+20{\log }_x(32)}}$ ?

6. In the diagram, $PABCD$ is a pyramid with square base $ABCD$ and with $PA=PB=PC=PD$. Suppose that $M$ is the midpoint of $PC$ and that $\mathrm{\angle }BMD={90}^{\circ }$. Triangular-based pyramid $MBCD$ is removed by cutting along the triangle defined by the points $M$, $B$ and $D$. The volume of the remaining solid $PABMD$ is 288. What is the length of $AB$?

   

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. 1. Factor $x^2-4$ as the product of two linear expressions in $x$.

   2. Determine the integer $k$ for which ${98}^2-4=100k$.

   3. Determine the positive integer $n$ for which $(20-n)(20+n)=391$.

   4. Prove that $3999991$ is not a prime number. (A prime number is a positive integer greater than 1 whose only positive divisors are 1 and itself. For example, 7 is a prime number.)

2. If $n$ is a positive integer, a Leistra sequence is a sequence $a_1,a_2,a_3,\dots ,a_{n-1},a_n$ with $n$ terms with the following properties:

   • (P1) Each term $a_1,a_2,a_3,\dots ,a_{n-1},a_n$ is an even positive integer.

   • (P2) Each term $a_2,a_3,\dots ,a_{n-1},a_n$ is obtained by dividing the previous term in the sequence by an integer between 10 and 50, inclusive. (For a specific sequence, the divisors used do not all have to be the same.)

   • (P3) There is no integer $m$ between 10 and 50, inclusive, for which ${\displaystyle \frac{a_n}{m}}$ is an even integer.

   For example,

   Leistra sequences
   1000, 50, 2
   1000, 100, 4
   3000, 300, 30, 2
   106

   Not Leistra sequences	Reason
   1000, 50, 1	(P1) fails – includes an odd integer
   1000, 200, 4	(P2) fails – divisor 5 falls outside range ($\frac{1000}{200}=5$)
   3000, 300, 30	(P3) fails – can be extended with $\frac{30}{15}=2$
   104	(P3) fails – can be extended with $\frac{104}{13}=8$

   1. Determine all Leistra sequences with $a_1=216$.

   2. How many Leistra sequences have $a_1=2\times 3^{50}$ ?

   3. How many Leistra sequences have $a_1=2^2\times 3^{50}$ ?

   4. Determine the number of Leistra sequences with $a_1=2^3\times 3^{50}$.

   (In parts (b) and (c), full marks will be given for a correct answer. Part marks may be awarded for an incomplete solution or work leading to an incorrect answer.)

3. A pair of functions $f(x)$ and $g(x)$ is called a Payneful pair if

   1. $f(x)$ is a real number for all real numbers $x$,

   2. $g(x)$ is a real number for all real numbers $x$,

   3. $f(x+y)=f(x)g(y)+g(x)f(y)$ for all real numbers $x$ and $y$,

   4. $g(x+y)=g(x)g(y)-f(x)f(y)$ for all real numbers $x$ and $y$, and

   5. $f(a)\ne 0$ for some real number $a$.

   For every Payneful pair of functions $f(x)$ and $g(x)$:

   1. Determine the values of $f(0)$ and $g(0)$.

   2. If $h(x)={(f(x))}^2+{(g(x))}^2$ for all real numbers $x$, determine the value of $h(5)h(-5)$.

   3. If $-10\le f(x)\le 10$ and $-10\le g(x)\le 10$ for all real numbers $x$, determine the value of $h(2021)$.