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Pascal Contest - Random Problems

Pascal Grade 9
March 1, 2026
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25
Problems
WaterlooMath
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13 hours ago
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Problems

  1. The expression $6a - 5a + 4a - 3a + 2a - a$ is equal to
    1. $3a$
    2. $3a^6$
    3. $3$
    4. $-21a$
    5. $-21a^6$
    (Source: 2005 Pascal Grade 9 , #2)

  2. When $x=9$, which of the following has the largest value?
    1. $\sqrt{x}$
    2. $\displaystyle\frac{x}{2}$
    3. $x-5$
    4. $\displaystyle\frac{40}{x}$
    5. $\displaystyle\frac{x^2}{20}$
    (Source: 2006 Pascal Grade 9 , #11)

  3. In the diagram, what is the area of $\triangle ABC$?
    The image depicts a triangle with three vertices labeled A, B, and C. The vertex A is located at (4, 9), vertex B is at (0, 0), and vertex C is at (12, 0). The x-axis is horizontal, and the y-axis is vertical.
    1. $36$
    2. $54$
    3. $108$
    4. $72$
    5. $48$
    (Source: 2007 Pascal Grade 9 , #8)

  4. In the diagram, $JLMR$ and $JKQR$ are rectangles. Also, $JR=2$, $RQ=3$ and $JL=8$. What is the area of rectangle $KLMQ$?
    The image depicts a simple diagram of two rectangles, labeled 'J' and 'L', with their corresponding lengths. The rectangle on the left is labeled 'J' and has a length of 2 units, while the rectangle on the right is labeled 'L' and has a length of 3 units. The width of both rectangles is not specified. The diagram appears to be a simple illustration of two different shapes, with no additional context or information provided. It does not seem to represent any specific concept or idea beyond its basic geometric properties.
    1. $6$
    2. $16$
    3. $10$
    4. $15$
    5. $24$
    (Source: 2008 Pascal Grade 9 , #4)

  5. The surface area of a large cube is 5400 cm$^2$. This cube is cut into a number of identical smaller cubes. Each smaller cube has a volume of 216 cm$^3$. How many smaller cubes are there?
    1. $25$
    2. $125$
    3. $164$
    4. $180$
    5. $216$
    (Source: 2008 Pascal Grade 9 , #17)

  6. If $10\%$ of $s$ is $t$, then $s$ equals
    1. $0.1t$
    2. $0.9t$
    3. $9t$
    4. $10t$
    5. $90t$
    (Source: 2010 Pascal Grade 9 , #15)

  7. John lists the integers from 1 to 20 in increasing order. He then erases the first half of the integers in the list and rewrites them in order at the end of the second half of the list. Which integer in the new list has exactly 12 integers to its left?
    1. $1$
    2. $2$
    3. $3$
    4. $12$
    5. $13$
    (Source: 2011 Pascal Grade 9 , #10)

  8. Two circles are centred at the origin, as shown. The point $P(8,6)$ is on the larger circle and the point $S(0,k)$ is on the smaller circle. If $QR=3$, what is the value of $k$?
    The image depicts a mathematical diagram, specifically a graph with two axes and several points plotted on it. The purpose of the image is to illustrate a geometric concept or relationship between these points. * A circle: + The circle is centered at point O. + It has a radius that extends from point O to point R. + Point S lies on the circumference of the circle, with coordinates (0, k). + Point P also lies on the circumference, with coordinates (8, 6). * Two axes: + The x-axis is horizontal and runs along the bottom of the image. + The y-axis is vertical and runs up the left side of the image. * Several points: + Point O is located at the origin (0, 0) where the two axes intersect. + Point R is located on the positive x-axis, with coordinates (8, 0). + Point S has coordinates (0, k), indicating that it lies directly above point O along the y-axis. + Point P has coordinates (8, 6), placing it in the first quadrant of the coordinate plane. The image shows a circle centered at point O, with points S and P lying on its circumference. The two axes provide context for understanding the coordinates of these points.
    1. $3.5$
    2. $4$
    3. $6$
    4. $6.5$
    5. $7$
    (Source: 2011 Pascal Grade 9 , #19)

  9. In the diagram, the horizontal distance between adjacent dots in the same row is 1. Also, the vertical distance between adjacent dots in the same column is 1. What is the perimeter of quadrilateral $PQRS$?
    The image depicts a graph with two axes, labeled 'P' and 'Q', which intersect at a point marked 'S'. The x-axis is not explicitly labeled but appears to be aligned with the horizontal axis. A line extends from point S to the right, intersecting the y-axis at point R. The y-axis is also unlabeled but seems to align with the vertical axis. The graph features several data points plotted as small circles, with one circle positioned directly above point S and another below it. These two points are connected by a line that extends downward from point S to the bottom of the graph. A third data point is located at the top-right corner of the graph, near point Q. A scale bar is visible along the x-axis, indicating a value of 1 unit. The background of the graph is white, providing a clean and neutral backdrop for the plotted data points and axes. Overall, the image presents a clear and concise visual representation of a mathematical concept or relationship between variables P and Q.
    1. $12$
    2. $13$
    3. $14$
    4. $15$
    5. $16$
    (Source: 2012 Pascal Grade 9 , #14)

  10. If $x=11$, $y=-8$, and $2x-3z=5y$, what is the value of $z$?
    1. $-6$
    2. $13$
    3. $54$
    4. $\frac{62}{3}$
    5. $-\frac{71}{3}$
    (Source: 2013 Pascal Grade 9 , #12)

  11. Sam rolls a fair four-sided die containing the numbers 1, 2, 3, and 4. Tyler rolls a fair six-sided die containing the numbers 1, 2, 3, 4, 5, and 6. What is the probability that Sam rolls a larger number than Tyler?
    1. $\frac{1}{8}$
    2. $\frac{5}{12}$
    3. $\frac{3}{5}$
    4. $\frac{3}{4}$
    5. $\frac{1}{4}$
    (Source: 2013 Pascal Grade 9 , #20)

  12. A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom, as shown. Each triangle is to be coloured one of red, white, blue, green, and purple so that no two triangles that share an edge are the same colour. How many different flags can be made?
    The image depicts a simple line drawing of a rectangle divided into four quadrants, with labels indicating the top-left, top-right, bottom-left, and bottom-right corners. The purpose of this image is to illustrate the concept of quadrants in a geometric context. * A rectangle is shown with its sides labeled. + The top side is labeled 'Top'. + The right side is labeled 'Right'. + The left side is labeled 'Left'. + The bottom side is labeled 'Bottom'. The image effectively conveys the idea of dividing a shape into four equal parts, each with its own distinct label. This visual representation can be useful for educational purposes or when explaining geometric concepts to others.
    1. $180$
    2. $200$
    3. $220$
    4. $240$
    5. $260$
    (Source: 2014 Pascal Grade 9 , #22)

  13. The expression $4 + \frac{3}{10} + \frac{9}{1000}$ is equal to
    1. $4.12$
    2. $4.309$
    3. $4.039$
    4. $4.012$
    5. $4.39$
    (Source: 2015 Pascal Grade 9 , #5)

  14. The average age of Andras, Frances and Gerta is 22 years. What is Gerta's age?
    1. $19$
    2. $20$
    3. $21$
    4. $22$
    5. $23$
    (Source: 2015 Pascal Grade 9 , #6)

  15. The first four rows of a table with columns $V$, $W$, $X$, $Y$, and $Z$ are shown. For each row, whenever integer $n$ appears in column $V$, column $W$ contains the integer $2n + 1$, column $X$ contains $3n + 1$, column $Y$ contains $5n+1$, and column $Z$ contains $7n+1$. For every row after the first, the number in column $V$ is the smallest positive integer that does not yet appear in any previous row. The integer 2731 appears in column $W$. The complete list of columns in which 2731 appears is
    The image presents a table with four columns labeled 'V', 'W', 'X', and 'Y' and three rows. The first row contains the numbers 1, 3, 4, and 6, while the second row has the numbers 2, 5, 7, and 11. The third row consists of the numbers 9, 19, 28, and 46. The fourth row is incomplete, with only the first two columns filled in: '10' and '21'.
    1. $W$
    2. $W$, $X$, $Y$, and $Z$
    3. $W$, $X$ and $Z$
    4. $W$, $Y$ and $Z$
    5. $W$ and $Z$
    (Source: 2015 Pascal Grade 9 , #25)

  16. A rectangle with height $x$ and width $2x$ has the same perimeter as an equilateral triangle with side length 10. What is the area of the rectangle? The image presents a simple geometric diagram, comprising two shapes: a rectangle and an equilateral triangle. The rectangle is positioned on the left side of the image, while the equilateral triangle is situated on the right. **Key Features:** * **Rectangle:** * Located on the left side of the image * Dimensions: 2 units wide and 1 unit tall (as indicated by the labels '2' and '1') * **Equilateral Triangle:** * Positioned on the right side of the image * Base length: 10 units (as labeled) * Height: Not explicitly stated, but can be inferred using the Pythagorean theorem or trigonometric ratios **Visual Representation:** The diagram is rendered in a clean and simple style, with black lines on a white background. The use of labels and dimensions provides clear information about the shapes' characteristics. **Conclusion:** In summary, the image depicts a rectangle and an equilateral triangle, highlighting their respective dimensions and relationships. This visual representation can be useful for educational or illustrative purposes in mathematics or geometry.
    1. 18
    2. 50
    3. 25
    4. 200
    5. 100
    (Source: 2018 Pascal Grade 9 , #13)

  17. In the diagram, $\triangle PQR$ has $\angle PQR = 120^\circ$. Also, $\angle QPS = \angle RPS$ and
    $\angle QRS = \angle PRS$. (In other words, $SP$ and $SR$ bisect $\angle QPR$ and $\angle QRP$, respectively.) What is the measure of $\angle PSR$? The image depicts a geometric diagram featuring two right-angled triangles, labeled 'PQR' and 'PSR', with their corresponding angles denoted as 90 degrees. The triangle PQR has an additional angle marked as 120 degrees. * **Triangle PQR:** * Angle PQR is 90 degrees. * Angle QPR is 60 degrees (calculated by subtracting the sum of the other two angles from 180 degrees). * Angle PRQ is also 30 degrees (calculated similarly). * **Triangle PSR:** * Angle PSR is 90 degrees. * Angle SQR is 120 degrees. The image provides a visual representation of these geometric relationships, allowing for the identification and calculation of various angles within the triangles.
    1. $130^\circ$
    2. $120^\circ$
    3. $140^\circ$
    4. $160^\circ$
    5. $150^\circ$
    (Source: 2018 Pascal Grade 9 , #18)

  18. If $2x + 6 = 16$, the value of $x+4$ is
    1. 7
    2. 8
    3. 9
    4. 15
    5. 13
    (Source: 2020 Pascal Grade 9 , #3)

  19. In the diagram, three lines intersect at a point. What is the value of $x$?
    The image depicts a geometric figure with two intersecting lines, forming an 'X' shape. The purpose of the image is to illustrate the concept of perpendicular lines. * A line: + The line is dark gray and extends from the top-left corner to the bottom-right corner. + It has a slight angle, making it not perfectly horizontal or vertical. * Another line: + This line is also dark gray and intersects with the first line at its midpoint. + It forms an 'X' shape with the first line, indicating that they are perpendicular to each other. * A symbol: + The symbol is a small circle with a dot in the center, located at the intersection point of the two lines. + It represents the concept of perpendicularity or orthogonality. The image effectively illustrates the idea of perpendicular lines and their relationship to each other.
    1. 30
    2. 45
    3. 60
    4. 90
    5. 120
    (Source: 2020 Pascal Grade 9 , #9)

  20. Ali, Bea, Che, and Deb compete in a checkers tournament. Each player plays each other player exactly once. At the end of each game, either the two players tie or one player wins and the other player loses. A player earns 5 points for a win, 0 points for a loss, and 2 points for a tie. Exactly how many of the following final point distributions are possible?
    \begin{array}{c|c} Player & Points \\ \hline Ali & 15 \\ Bea & 7 \\ Che & 4 \\ Deb & 2 \end{array} \begin{array}{c|c} Player & Points \\ \hline Ali & 10 \\ Bea & 10 \\ Che & 4 \\ Deb & 4 \end{array} \begin{array}{c|c} Player & Points \\ \hline Ali & 15 \\ Bea & 5 \\ Che & 5 \\ Deb & 2 \end{array} \begin{array}{c|c} Player & Points \\ \hline Ali & 12 \\ Bea & 10 \\ Che & 5 \\ Deb & 0 \end{array}
    1. 0
    2. 1
    3. 2
    4. 3
    5. 4
    (Source: 2020 Pascal Grade 9 , #23)

  21. The expression \(\dfrac{20+22}{2}\) is equal to
    1. \(1\)
    2. \(4\)
    3. \(20\)
    4. \(21\)
    5. \(22\)
    (Source: 2022 Pascal Grade 9 , #1)

  22. Alvin, Bingyi and Cheska play a two-player game that never ends in a tie. In a recent tournament between the three players, a total of 60 games were played and each pair of players played the same number of games. How many games did Bingyi win?
    1. \(12\)
    2. \(24\)
    3. \(28\)
    4. \(30\)
    5. \(36\)
    (Source: 2022 Pascal Grade 9 , #16)

  23. Jurgen is travelling to Waterloo by bus. He packs for 25 minutes. He then walks to the bus station, which takes 35 minutes. He arrives 60 minutes before his bus leaves. His bus leaves at 6:45 p.m. At what time did he start packing?
    1. 4:45 p.m.
    2. 4:40 p.m.
    3. 4:35 p.m.
    4. 4:55 p.m.
    5. 4:50 p.m.
    (Source: 2023 Pascal Grade 9 , #7)

  24. A Pretti number is a seven-digit positive integer with the following properties:

    • The integer formed by its leftmost three digits is a perfect square.

    • The integer formed by its rightmost four digits is a perfect cube.

    • Its ten thousands digit and ones (units) digit are equal.

    • Its thousands digit is not zero.

    How many Pretti numbers are there?

    (Source: 2022 Pascal Grade 9 , #24)

  25. A \(3 \times 3\) table starts with every entry equal to \(0\) and is modified using the following steps:
    (i) adding \(1\) to all three numbers in any row;
    (ii) adding \(2\) to all three numbers in any column.

    After step (i) has been used a total of \(a\) times and step (ii) has been used a total of \(b\) times, the table appears as shown.

    \(7\) \(1\) \(5\)
    \(9\) \(3\) \(7\)
    \(8\) \(2\) \(6\)

    What is the value of \(a+b\)?

    (Source: 2024 Pascal Grade 9 , #21)

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