Maths Olympiad Prep

For teachers / Printable sets /Stage 2 · Combinatorics

Stage 2 · Combinatorics

10 problems · · mathsolympiadprep.com

The answer key prints on its own page at the end.

  1. In a class, 3030 students did a test. Every student got a grade that was an integer from 11 to 1010. The grade 88 was given more often than any of the other grades.
    What is the smallest possible average grade of the students?
    A) 38153\frac{8}{15}
    B) 3233\frac{2}{3}
    C) 3563\frac{5}{6}
    D) 411304\frac{11}{30}
    E) 48154\frac{8}{15}

    Combinatorics Solution and answer checking →

  2. In a group of five friends, Amy is taller than Carla. Dan is shorter than Eric but taller than Bob. Eric is shorter than Carla. Who is the shortest?

    Combinatorics Solution and answer checking →

  3. In the diagram, the target shown has three scoring areas. An
    arrow that hits the centre circle is worth 1010 points, an arrow that hits the shaded
    middle ring is worth 55 points, and
    an arrow that hits the outer ring is worth 11 point.

    Three arrows are shot and each hits the target. Which of the
    following cannot be the total score for the three arrows?

    1. A1616
    2. B1111
    3. C1313
    4. D77
    5. E2020

    Combinatorics Solution and answer checking →

  4. A soccer team played three games. Each game ended in a win, loss, or tie. (If a game finishes with both teams having scored the same number of goals, the game ends in a tie.) In total, the team scored more goals than were scored against them. Which of the following combinations of outcomes is not possible for this team?

    2 wins, 0 losses, 1 tie\text{2 wins, 0 losses, 1 tie}
    1 win, 2 losses, 0 ties\text{1 win, 2 losses, 0 ties}
    0 wins, 1 loss, 2 ties\text{0 wins, 1 loss, 2 ties}
    1 win, 1 loss, 1 tie\text{1 win, 1 loss, 1 tie}
    1 win, 0 losses, 2 ties\text{1 win, 0 losses, 2 ties}

    Combinatorics Solution and answer checking →

  5. At Barker High School, a total of 36 students are on either the baseball team, the hockey team, or both. If there are 25 students on the baseball team and 19 students on the hockey team, how many students play both sports?

    1. A77
    2. B88
    3. C99
    4. D1010
    5. E1111

    Combinatorics Solution and answer checking →

  6. Shaded and unshaded squares are arranged in rows so that:

    the first row consists of one unshaded square,
    each row begins with an unshaded square,
    the squares in each row alternate between unshaded and shaded, and
    each row after the first has two more squares than the previous row.

    The first 4 rows are shown.

    The number of shaded squares in the 2020th row is

    20222022
    20212021
    20202020
    20192019
    20182018

    Combinatorics Solution and answer checking →

  7. The two equal-arm scales shown are balanced.

    Which of the following has the same mass as





    Combinatorics Solution and answer checking →

  8. On the map shown, there are a number of routes from Mathville to
    Algebratown.

    Each route must travel along the roads in the direction marked by the
    arrows. The total number of routes from Mathville to Algebratown is

    1. A33
    2. B44
    3. C88
    4. D66
    5. E1010

    Combinatorics Solution and answer checking →

  9. Twelve lightbulbs are in a row. All lightbulbs are initially
    turned off. Angie flips the switch for every 2nd lightbulb. Then Bilal
    flips the switch for every 3rd lightbulb. Finally, Chenxhui flips the
    switch for every 4th lightbulb. At the end of this process, how many
    lightbulbs are turned on?

    1. A55
    2. B66
    3. C77
    4. D88
    5. E99

    Combinatorics Solution and answer checking →

  10. Two cubes are stacked as shown. The faces of each cube are labelled with 1, 2, 3, 4, 5, and 6 dots. A total of five faces are shown.

    What is the total number of dots on the other seven faces of these two cubes?

    1. A1313
    2. B1414
    3. C1818
    4. D2121
    5. E2424

    Combinatorics Solution and answer checking →

Answer key — Stage 2 · Combinatorics

Worked solutions for every problem are on the site, one page per problem.

  1. BB open
  2. BobBob open
  3. CC open
  4. DD open
  5. BB open
  6. DD open
  7. EE open
  8. CC open
  9. AA open
  10. EE open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.