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2022 Canadian Intermediate
Mathematics Contest

Wednesday, November 16, 2022
(in North America and South America)

Thursday, November 17, 2022
(outside of North American and South America)

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©2022 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

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. The area of one face of a cube is \(16\mbox{ cm}^2\). The volume of the same cube is \(V\mbox{ cm}^3\). What is the value of \(V\)?

  2. In the diagram, point \(D\) is on side \(BC\) of \(\triangle ABC\). If \(BD=CD=AD\) and \(\angle ACD = 40^\circ\), what is the measure of \(\angle BAC\)?

  3. Marie-Pascale solves 4 math problems per day. Kaeli solves \(x\) math problems per day. Over a number of days, Marie-Pascale solves a total of 72 problems. Over the same number of days, Kaeli solves 54 more problems than Marie-Pascale. What is the value of \(x\)?

  4. If \(a\), \(b\), \(c\), and \(d\) satisfy \(\dfrac{a}{b} = \dfrac{2}{3}\) and \(\dfrac{c}{b} =\dfrac{1}{5}\) and \(\dfrac{c}{d} = \dfrac{7}{15}\), what is the value of \(\dfrac{ab}{cd}\)?

  5. Magnus and Viswanathan play a game against each other three times:

    What was Magnus’s score in the third game?

  6. How many ways are there to choose integers \(a\), \(b\) and \(c\) with \(a < b < c\) from the list 1, 5, 8, 21, 22, 27, 30, 33, 37, 39, 46, 50 so that the product \(abc\) is a multiple of 12?

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 each part of this problem, seven different positive integers will be placed in the seven boxes of the "H"-shaped figure.

    The left vertical side of a capital H is formed by three boxes placed in a vertical column, as is the right vertical side of the H. A middle box is placed between the vertical columns to form the horizontal row of the H. From top to bottom, the left column has the numbers 8, 11, and 12, and the right column has 27, 1, and 3. The middle box has 19.

    The integers must be placed so that the three integers in the left vertical column, the three integers in the right vertical column, and the three integers in the one horizontal row all have the same sum. For example, in the given "H", \(8+11+12=31\) and \(11+19+1=31\) and \(27+1+3=31\).

    1. Place the integers 3, 5, 7, 15 in the figure so that each of the three sums is equal to 29.

      The left vertical column has 13 in the top box and 9 in the next box. The right vertical column has 11 in the top box. The other four boxes are empty.

    2. There is a value of \(t\) for which the figure shown has three equal sums and contains seven different integers. Determine this value of \(t\).

      From top to bottom, the left column has the number 15, the expression t plus 1, and the number 11. The right column has the expression 2 times t minus 3, the number 10, and the number 14. The middle box has the number 16.

    3. Seven different positive integers are placed in the figure so that the three sums are equal.

      From top to bottom, the left column has the variable a, the numbers 12, and the variable c. The right column has the variable b, the number 7, and the number 11. The middle box has the number 9.

      If \(a < c\), determine all possible values of \(a\).

    4. The integers \(k\) and \(n\) are each between 4 and 18, inclusive. The figure contains seven different integers and the three sums are equal.

      From top to bottom, the left column has the number 7, the number 10, and the variable k. The right column has the expression n plus 6, the number 18, and the number 4. The middle box has the variable n.

      Determine all possible values of \(k\).

  2. A line that is neither horizontal nor vertical intersects the \(y\)-axis when \(x=0\) and intersects the \(x\)-axis when \(y=0\). For example, the line with equation \(4x + 5y = 40\) intersects the \(y\)-axis at \((0,8)\) (because \(4\times 0 + 5y = 40\) gives \(y = 8\)) and the \(x\)-axis at \((10,0)\) (because \(4x + 5 \times 0 = 40\) gives \(x = 10\)).

    1. The line with equation \(2x + 3y = 12\) intersects the \(y\)-axis at \(A\) and the \(x\)-axis at \(B\). If \(O\) is the origin, determine the area of \(\triangle AOB\).

    2. Suppose that \(c>0\). The line with equation \(6x + 5y = c\) intersects the \(y\)-axis at \(D\) and the \(x\)-axis at \(E\). If \(O\) is the origin and the area of \(\triangle DOE\) is 240, determine the value of \(c\).

    3. Suppose that \(m\) and \(n\) are integers with \(100 \leq m\) and \(m < n\). The line with equation \((2m)x + y = 4m\) intersects the \(y\)-axis at \(P\) and the \(x\)-axis at \(Q\). The line with equation \((7n)x + 4y = 28n\) intersects the \(y\)-axis at \(S\) and the \(x\)-axis at \(R\). If quadrilateral \(PQRS\) has area 2022, determine two possible pairs \((m,n)\).

  3. A straight path is 2 km in length. Beatrice walks from the beginning of the path to the end of the path at a constant speed of 5 km/h. Hieu cycles from the beginning of the path to the end of the path at a constant speed of 15 km/h.

    1. Suppose that Hieu starts 10 minutes later than Beatrice. Determine the number of minutes that it takes Hieu to catch up to Beatrice on the path.

    2. Suppose that Beatrice starts at a time that is exactly \(b\) minutes after 9:00 a.m. and that Hieu starts at a time that is exactly \(h\) minutes after 9:00 a.m., where \(b\) and \(h\) are integers from 0 to 59, inclusive, that are each randomly and independently chosen.

      1. Determine the probability that there is a time at which Beatrice and Hieu are at the same place on the path. (Beatrice and Hieu are at the same place at a given time if they start at the same time or finish at the same time, or are at the same place at some point between the beginning and end of the path at the same time.)

      2. One day, Beatrice uses a scooter and travels from the beginning of the path to the end of the path at a constant speed of \(x\) km/h, where \(x > 5\) and \(x < 15\). Hieu still cycles from the beginning of the path to the end of the path at a constant speed of 15 km/h. If the probability that there is a time at which Beatrice and Hieu are at the same place on the path is \(\frac{13}{200}\), determine the range of possible values of \(x\).