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2026 Pascal Contest
(Grade 9)

Wednesday, February 25, 2026
(in North America and South America)

Thursday, February 26, 2026
(outside of North American and South America)

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©2026 University of Waterloo

Instructions

Time: 60 minutes

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.

  1. Do not open the Contest booklet until you are told to do so.
  2. You may use rulers, compasses and paper for rough work.
  3. Be sure that you understand the coding system for your response form. If you are not sure, ask your teacher to clarify it. All coding must be done with a pencil, preferably HB. Fill in circles completely.
  4. On your response form, print your school name and city/town in the box in the upper right corner.
  5. Be certain that you code your name, age, grade, and the Contest you are writing in the response form. Only those who do so can be counted as eligible students.
  6. Part A and Part B of this contest are multiple choice. Each of the questions in these parts is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. After making your choice, fill in the appropriate circle on the response form.
  7. The correct answer to each question in Part C is an integer from 0 to 99, inclusive. After deciding on your answer, fill in the appropriate two circles on the response form. A one-digit answer (such as "7") must be coded with a leading zero ("07").
  8. Scoring:
    1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
    2. There is no penalty for an incorrect answer.
    3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
  9. Diagrams are not drawn to scale. They are intended as aids only.
  10. When your supervisor tells you to begin, you will have sixty minutes of working time.
  11. You may not write more than one of the Pascal, Cayley and Fermat Contests in any given year.
Do not discuss the problems or solutions from this contest online for the next 48 hours.
The name, grade, school and location, and score range of some top-scoring students will be published on the website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some students may be shared with other mathematical organizations for other recognition opportunities.
Scoring:
  1. There is no penalty for an incorrect answer.
  2. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth 5.

  1. The value of \(8-7+6-5+4-3+2-1\) is

    1. \(4\)
    2. \(6\)
    3. \(5\)
    4. \(8\)
    5. \(3\)

  2. The value of \(\sqrt{37}\) is closest to

    1. \(5\)
    2. \(6\)
    3. \(7\)
    4. \(18\)
    5. \(74\)

  3. In the equation \(2+2+2+2=2^x\), what is the value of \(x\)?

    1. \(6\)
    2. \(4\)
    3. \(3\)
    4. \(5\)
    5. \(2\)

  4. If the numbers \(2\frac{3}{4}\), 2.3, \(2\frac{1}{4}\), 2.45, and \(2\frac{9}{10}\) were arranged from least to greatest, the middle number would be

    1. \(2\frac{3}{4}\)
    2. \(2.45\)
    3. \(2\frac{1}{4}\)
    4. \(2.3\)
    5. \(2\frac{9}{10}\)

  5. In the diagram, \(9\) edges of the cube are visible.

    A cube with front face, top face, and right face visible.

    How many edges of the cube are not visible?

    1. \(0\)
    2. \(1\)
    3. \(2\)
    4. \(3\)
    5. \(4\)

  6. Five paths from \(A\) to \(B\) are shown. In which figure is the path the shortest?

    The path moves 4 units up then 4 units to the right to the right.
    Figure 1
    The path moves 2 units to the right, then 2 units up, then 2 units to the right, then 2 units up.
    Figure 2
    The path moves 2 units up, then 1 unit to the right, then 1 unit up, then 3 units to the right, then 1 unit up.
    Figure 3
    The path moves 2 units to the right, then along the diagonal of a 2 by 2 square, then 2 units up.
    Figure 4
    The path moves 1 unit to the right, then 1 unit up, then 1 unit to the right, then 1 unit up, then 1 unit to the right, then 1 unit up, then 1 unit to the right, then 1 unit up.
    Figure 5
    1. Figure 1
    2. Figure 2
    3. Figure 3
    4. Figure 4
    5. Figure 5

  7. A train consists of one engine and \(n\) freight cars. The engine’s length is \(20\) m. Each freight car has length \(15\) m. If the combined length of the engine and the freight cars is \(140\) m, then \(n\) equals

    1. \(5\)
    2. \(8\)
    3. \(7\)
    4. \(6\)
    5. \(9\)

  8. Which of the following integers has the same remainder when it is divided by \(3\) as when it is divided by \(4\)?

    1. \(9\)
    2. \(11\)
    3. \(10\)
    4. \(13\)
    5. \(8\)

  9. In the diagram, \(\triangle PQT\) is equilateral. Both \(\triangle QST\) and \(\triangle QRS\) are right-angled triangles and \(QR=RS\).

    Triangle QST has hypotenuse QT and right angle at S. Triangle QRS has hypotenuse QS and right angle at R.

    If \(\angle STP=120\degree\), the measure of \(\angle PQR\) is

    1. \(145\degree\)
    2. \(105\degree\)
    3. \(90\degree\)
    4. \(120\degree\)
    5. \(135\degree\)

  10. In the diagram, six \(4\times 4\) squares have a percentage of their area shaded.

    A description of the diagram follows.

    How many of the \(4\times 4\) squares have exactly \(25\%\) of their area shaded?

    1. \(2\)
    2. \(4\)
    3. \(3\)
    4. \(6\)
    5. \(5\)

Part B: Each correct answer is worth 6.

  1. In the diagram, the number placed in each rectangle must equal the sum of the numbers in the two rectangles directly below it. For example, the rectangles containing \(1\) and \(3\) are directly below the rectangle containing \(x\), and so \(x=1+3=4\). What is the value of \(y\)?

    11 identical rectangles arranged into 3 rows. The bottom row has 5 rectangles. From left to right, they have numbers 9, 2, empty, 1, 3. The middle row has 4 rectangles and is centred so each rectangle has two rectangles directly below it. The rightmost rectangle in this row contains x. A top rectangle containing 8 has the two leftmost rectangles in the middle row directly below it. Another top rectangle containing y has the two rightmost rectangles in the middle row directly below it.

    1. \(7\)
    2. \(8\)
    3. \(9\)
    4. \(10\)
    5. \(11\)

  2. In the diagram, the point \(P(2, 3)\) is reflected in the \(x\)-axis and then moved \(3\) units left. What are the coordinates of the resulting point?

    Point P plotted on the xy-plane with x and y axes both ranging from negative 5 to 5.

    1. \((-1,-3)\)
    2. \((-5,3)\)
    3. \((1,3)\)
    4. \((5,-3)\)
    5. \((2,-6)\)

  3. Max was given \(b\) books. In March, he read one third of these books. In April, he read \(5\) more of the books. Max then had \(7\) books left to read. If Max never read the same book more than once, what is the value of \(b\)?

    1. \(18\)
    2. \(15\)
    3. \(27\)
    4. \(21\)
    5. \(24\)

  4. In the diagram, square \(WXYZ\) has side length \(15\). Point \(V\) is placed on \(WX\) so that \(VZ = 17\). What is the area of trapezoid \(VXYZ\)?

    1. \(97.5\)
    2. \(157\)
    3. \(161.5\)
    4. \(165\)
    5. \(229\)

  5. Peter’s car uses \(10.2\) L of fuel per \(100\) km. Mike’s hybrid car uses \(6.6\) L of fuel per \(100\) km. Fuel costs \(\$1.40\) per litre. If both Peter and Mike drive \(200\) km, how much less does Mike spend on fuel?

    1. \(\$3.96\)
    2. \(\$5.32\)
    3. \(\$7.36\)
    4. \(\$8.63\)
    5. \(\$10.08\)

  6. \(P\), \(Q\), \(R\), and \(S\) are four distinct points on a line segment in the order shown.

    From left to right, the points on the segment are P, Q, R, then S.

    If \(PR=8\) and \(QS=15\), what is the smallest possible integer length of \(PS\)?

    1. \(22\)
    2. \(21\)
    3. \(18\)
    4. \(16\)
    5. \(23\)

  7. Four of the five puzzle pieces shown can fit together without overlap to form a square.

    A description of the diagram follows.

    If the pieces can be rotated but not flipped, which piece does not get used?

    1. Piece 1
    2. Piece 2
    3. Piece 3
    4. Piece 4
    5. Piece 5

  8. When the integers from \(17\) to \(352\) are added, the sum is \[17+18+19+\cdots+350+351+352=61\,992\] When the integers from \(20\) to \(355\) are added, the sum is\[20+21+22+\cdots+353+354+355=x\] What is the value of \(x\)?

    1. \(61\,995\)
    2. \(63\,003\)
    3. \(62\,040\)
    4. \(62\,997\)
    5. \(63\,000\)

  9. The diagram consists of thirty-four \(1\times 1\) squares. Using only the grid lines to form squares, how many squares of all sizes are in the diagram?

    The bottom of the figure is grid of squares with three rows and six columns. Placed along the top side of this grid, aligning with the middle four columns, is a four by four grid of squares.

    1. \(71\)
    2. \(70\)
    3. \(72\)
    4. \(69\)
    5. \(68\)

  10. A bag contains two quarters (worth \(\$0.25\) each), two dimes (worth \(\$0.10\) each), and two nickels (worth \(\$0.05\) each). Two coins are randomly chosen from the bag. Each coin is equally likely to be chosen. The probability that the combined value of the two coins is \(\$0.30\) or more is

    1. \(\dfrac{8}{15}\)
    2. \(\dfrac 35\)
    3. \(\dfrac 59\)
    4. \(\dfrac 13\)
    5. \(\dfrac 45\)

Part C: Each correct answer is worth 8.

Each correct answer is an integer from 0 to 99, inclusive.

  1. A lock requires a three-digit combination with the following characteristics:

    How many possible lock combinations have these characteristics?

  2. Consider the five distinct integers \(16\), \(x\), \(8\), \(17\), and \(11\). Their mean (average) and their median are equal. What is the sum of all possible values of \(x\)?

  3. On each side of the square shown, a semi-circle is drawn inside the square. The side length of the square is \(10\) and is equal to the diameter of each semi-circle. The four semi-circles overlap to form the shaded four-petal flower.

    Each of the four petals extends from the centre of the square to one of the four vertices of the square. The four petals are shaded.

    If \(n\) is the closest integer to the area of the shaded flower, what is the value of \(n\)?

  4. The string of digits \(123451234551234555\ldots\) is formed by alternately writing the digits \(1234\), in that order, and then writing some number of consecutive \(5\)s. There are exactly \(k\) consecutive \(5\)s immediately following the \(k\)th occurrence of \(1234\). If \(S\) is the sum of the first \(2026\) digits of the string, what is the sum of the digits of \(S\)?

  5. Triangle \(ABC\) is equilateral with side length \(6\), as shown in Figure 1. Three smaller equilateral triangles with positive integer side lengths \(a\leq b\leq c\) are removed from the corners of \(\triangle ABC\), as in Figure 2. If \(a+b<6\), \(a+c<6\), and \(b+c<6\), then the resulting figure, \(PQRSTU\), is a hexagon, as in Figure 3.

    Figure 1
    Figure 2
    Figure 3

    \(PQRSTU\) can be divided into exactly \(n\) identical equilateral triangles, each with a positive integer side length. If \(M\) is the sum of all possible values of \(n\), then what are the rightmost two digits of \(M\)?

Further Information

For students...

Thank you for writing the Pascal Contest!

Encourage your teacher to register you for the Fryer Contest which will be written in April.

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to