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2026 Gauss Contest
Grade 8

May 11 to May 22, 2026
(in North America and South America)

May 11 to May 22, 2026
(outside of North American and South America)

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©016 University of Waterloo

Instructions

Time: 1 hour

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 answer sheet. If you are not sure, ask your teacher to explain it.
  4. This is a multiple-choice test. Each question is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. When you have made your choice, enter the appropriate letter for that question on your answer sheet.
  5. 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.
  6. Diagrams are not drawn to scale. They are intended as aids only.
  7. When your supervisor instructs you to start, you will have sixty minutes of working time.
The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation.
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 mode of the list of numbers \(2\), \(8\), \(2\), \(5\), \(2\), \(3\), \(2\) is

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

  2. Liliana owes Abigail \(\$11\). If she gives Abigail \(\$4\), how much does she still owe?

    1. \(\$6\)
    2. \(\$4\)
    3. \(\$11\)
    4. \(\$7\)
    5. \(\$8\)

  3. A triangle has side lengths \(3~\text{cm}\), \(5~\text{cm}\), and \(x~\text{cm}\), as shown.

    If the perimeter of the triangle is \(12~\text{cm}\), the value of \(x\) is

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

  4. A leaking faucet drips at a rate of \(1\) drop every \(10\) seconds. How many drops will fall in \(1\) minute?

    1. \(100\)
    2. \(10\)
    3. \(15\)
    4. \(45\)
    5. \(6\)

  5. In the diagram, the number line between \(0\) and \(4\) is divided into \(16\) equal parts. The number \(T\) is marked on the line. What is the value of \(T\)?

    Tick marks divide the number line between 0 and 4 as described. The number T is 3 tick marks to the right of 0.

    1. \(0.8\)
    2. \(1.25\)
    3. \(0.75\)
    4. \(3\)
    5. \(0.5\)

  6. Quadrilateral \(RSTU\) is formed by placing two equilateral triangles together, as shown.

    Congruent equilateral triangles RSU and TSU share side SU.

    The measure of \(\angle RST\) is

    1. \(60\degree\)
    2. \(90\degree\)
    3. \(100\degree\)
    4. \(120\degree\)
    5. \(150\degree\)

  7. A grade 8 class surveyed its students to see how they get to school each day. The data collected is in the chart shown.

    How student
    gets to school    		 Number of
    students
    Walk \(10\)
    Bike \(5\)
    Bus \(9\)
    Car \(6\)

    The percentage of the students that get to school by car is

    1. \(4\%\)
    2. \(6\%\)
    3. \(20\%\)
    4. \(30\%\)
    5. \(25\%\)

  8. Livy must enter a four-digit code to unlock her phone. She knows that all four digits are different and that each digit is an integer from \(0\) to \(9\) inclusive. If she remembers the first three digits, but forgets the last digit, what is the probability that she will enter the correct code on her first try?

    1. \(\dfrac{1}{9}\)
    2. \(\dfrac{1}{7}\)
    3. \(\dfrac{7}{10}\)
    4. \(\dfrac{1}{6}\)
    5. \(\dfrac{1}{4}\)

  9. In the diagram, a shaded rectangle is drawn on each of four faces of a cube to form a band around the cube.

    Each of the four vertical sides of the cube are divided into three horizontal strips. The middle strip is shaded on each face, forming a band around the cube.

    Which of the following could not be the net of the cube?

    1. 4 squares form the middle row of the net. The 2nd and 4th of these squares are each divided into 3 vertical strips with the middle strip shaded. There are also squares above and below the 1st square. These additional squares are each divided into 3 horizontal strips with the middle strip shaded.
    2. 4 squares form the middle row. The 2nd and 4th are divided into three vertical strips with the middle strip shaded. There is also a square above the 3rd square and below the 4th square. These additional squares are divided into 3 vertical strips with the middle strip shaded.
    3. 4 squares form the middle row. All 4 are divided into 3 horizontal strips with the middle strip shaded. There is also a square above the 2nd square and below the 4th square.
    4. 4 squares form the middle row. The 1st and 3rd are divided into 3 vertical strips with the middle strip shaded. There are also squares above and below the 2nd square. These additional squares are divided into 3 horizontal strips with the middle strip shaded.
    5. 4 squares form the middle row. The 2nd and 4th are divided into 3 vertical strips with the middle strip shaded. There is also a square above the 2nd square and it is divided into 3 vertical strips with the middle strip shaded. There is also a square below the 3rd square and it is divided into 3 horizontal strips with the middle strip shaded.

  10. Fatima writes the list of positive integers in order, \(1, 2, 3, 4, \ldots\) and so on. The 15th positive even integer in the list is subtracted from the 25th positive odd integer in the list. The result is

    1. \(15\)
    2. \(17\)
    3. \(19\)
    4. \(21\)
    5. \(23\)

Part B: Each correct answer is worth 6.

  1. If \(p+q+r=19\) and \(p+r=8\), then the value of \(q\) is

    1. \(10\)
    2. \(11\)
    3. \(12\)
    4. \(13\)
    5. \(14\)

  2. Abel, Brock, Callie, Duan, and Edith are sitting around a circular table. Brock sits in the chair between Abel and Duan. Edith is not beside Duan. Who is sitting to Edith's immediate left and right?

    1. Abel and Callie
    2. Brock and Duan
    3. Abel and Brock
    4. Brock and Callie
    5. Callie and Duan

  3. The mass of a serving tray is \(4\) times the mass of a plate. A waiter is holding a serving tray with a stack of \(6\) plates on the tray. If the combined mass of the tray and the plates is \(3000~\text{g}\), what is the mass of each plate?

    1. \(600~\text{g}\)
    2. \(150~\text{g}\)
    3. \(30~\text{g}\)
    4. \(500~\text{g}\)
    5. \(300~\text{g}\)

  4. Anna and Yao are \(42~\text{km}\) apart. They jog towards each other along a straight road. Anna jogs at a constant rate of \(6~\text{km/h}\) and Yao jogs at a constant rate of \(8~\text{km/h}\). When they meet, how much farther has Yao travelled than Anna?

    1. \(6~\text{km}\)
    2. \(2~\text{km}\)
    3. \(4~\text{km}\)
    4. \(21~\text{km}\)
    5. \(24~\text{km}\)

  5. The height of a stack of identical newspapers is \(50~\text{cm}\). Half of the newspapers are removed from the stack. Then \(\frac{1}{5}\) of those removed are put back onto the stack. What is the height of the stack now?

    1. \(10~\text{cm}\)
    2. \(25~\text{cm}\)
    3. \(30~\text{cm}\)
    4. \(35~\text{cm}\)
    5. \(40~\text{cm}\)

  6. In the diagram, the right-angled isosceles triangle has a hypotenuse length of \(\sqrt{8}~\text{cm}\).

    The two legs of the triangle each have length x cm.

    What is the smallest number of these triangles needed to completely cover a square with side length \(8~\text{cm}\)?

    1. \(4\)
    2. \(8\)
    3. \(16\)
    4. \(32\)
    5. \(64\)

  7. Each of \(P4R\), \(7QS\) and \(TU1\) is a positive three-digit integer. The sum of \(P4R\) and \(7QS\) is equal to \(TU1\), as shown. \[\begin{array}{ccccc} &&\!\!\!P\!\!\!&\!\!\!4\!\!\!&\!\!\!R\!\!\! \\ + &&\!\!\!7\!\!\!&\!\!\!Q\!\!\!&\!\!\!S\!\!\! \\ \hline &&\!\!\!T\!\!\!&\!\!\!U\!\!\!&\!\!\!1\!\!\! \end{array}\] If each of \(P\), \(Q\), \(R\), \(S\), \(T\), and \(U\) represents a different digit from the list \(0\), \(2\), \(3\), \(5\), \(8\), \(9\), what is the value of \(U\)?

    1. \(0\)
    2. \(2\)
    3. \(3\)
    4. \(5\)
    5. \(8\)

  8. The Dollop sells ice cream cones. Each cone sold is exactly one of three different types: chocolate or vanilla or twist (chocolate and vanilla combined). Of all cones sold, \(\frac{11}{17}\) contained some chocolate and \(\frac{10}{17}\) contained some vanilla. If \(85\) ice cream cones were sold in total, how many were twist cones?

    1. \(15\)
    2. \(4\)
    3. \(25\)
    4. \(17\)
    5. \(20\)

  9. The line passing through the points \(A\), \(B\), \(C\), and \(D\), is parallel to the line passing through the points \(E\), \(F\), and \(G\), as shown.

    \(ABFE\) is a parallelogram and \(CDG\) is a triangle. If \(AB = 8~\text{cm}\) and \(CD = 12~\text{cm}\), what is the ratio of the area of \(ABFE\) to the area of \(\triangle CDG\)?

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

  10. If each of \(a\), \(b\) and \(c\) is a positive integer so that \(a+\dfrac{1}{b+\frac 1c}=\dfrac{90}{11}\), then what is the value of \(a+b+c\) ?

    1. \(16\)
    2. \(14\)
    3. \(21\)
    4. \(15\)
    5. \(13\)

Part C: Each correct answer is worth 8.

  1. How many possible ways can \(r\), \(s\) and \(t\) each be assigned a different number from the list \(3\), \(4\), \(5\) so that the value of \(r\times s + t\) is odd?

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

  2. In the diagram, points \(A\), \(B\), \(C\) are on a circle with centre \(D\) and radius \(5~\text{cm}\) so that \(AB=4~\text{cm}\) and \(BC=6~\text{cm}\). The points \(M\) and \(N\) are the midpoints of \(AB\) and \(BC\), respectively.

    Rounded to one decimal place, the area of \(DMBN\) is

    1. \(\text{21.2 cm}^2\)
    2. \(\text{10.6 cm}^2\)
    3. \(\text{27.0 cm}^2\)
    4. \(\text{11.4 cm}^2\)
    5. \(\text{9.8 cm}^2\)

  3. A die is created with the integers \(1\), \(2\), \(2\), \(4\), \(7\), \(8\) on its six faces. The first time the die is rolled, the numbers on its faces are changed in the following way:

    The die is then rolled a second time. What is the probability that the second number rolled is \(2\)?

    1. \(\dfrac{1}{9}\)
    2. \(\dfrac{2}{3}\)
    3. \(\dfrac{1}{3}\)
    4. \(\dfrac{5}{18}\)
    5. \(\dfrac{1}{6}\)

  4. Serafine and Jamie are playing catch with special rules. They start by standing \(1500~\text{cm}\) apart. One throws the ball to the other. If the ball is caught, both players each take a \(60~\text{cm}\) step away from each other. If the ball is not caught, the person who missed the catch takes a \(20~\text{cm}\) step towards the other player. When they stop playing catch, they are \(1720~\text{cm}\) apart. Of the following, which number of throws is not possible?

    1. \(17\)
    2. \(21\)
    3. \(73\)
    4. \(45\)
    5. \(10\)

  5. Five distinct points are located on a line. There are ten distances between pairs of points. Nine of the ten distances are \(2\), \(3\), \(4\), \(5\), \(7\), \(9\), \(11\), \(13\), \(16\). The remaining distance is \(n\), and it is possible that \(n\) is equal to one of the other nine distances. If \(n\) is a positive integer from \(1\) to \(15\) inclusive, how many values of \(n\) are possible?

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