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Stage 2 · Combinatorics

10 problems · · mathsolympiadprep.com

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1. In a class, $30$ students did a test. Every student got a grade that was an integer from $1$ to $10$. The grade $8$ was given more often than any of the other grades.
   What is the smallest possible average grade of the students?
   A) $3\frac{8}{15}$
   B) $3\frac{2}{3}$
   C) $3\frac{5}{6}$
   D) $4\frac{11}{30}$
   E) $4\frac{8}{15}$

   Junior Mathematical Olympiad, September 2019

   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?

   CEMC Cayley

   Combinatorics Solution and answer checking →

3. 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?

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

   CEMC Pascal 2016

   Combinatorics Solution and answer checking →

4. 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. A$7$
   2. B$8$
   3. C$9$
   4. D$10$
   5. E$11$

   CEMC Fermat 2015

   Combinatorics Solution and answer checking →

5. Each of five doors is randomly either open or closed. What is the
   probability that exactly two of the five doors are open?

   1. A$\frac{5}{16}$
   2. B$\frac{13}{32}$
   3. C$\frac{9}{25}$
   4. D$\frac{1}{2}$
   5. E$\frac{7}{16}$

   CEMC Cayley 2023

   Combinatorics Solution and answer checking →

6. 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. A$5$
   2. B$6$
   3. C$7$
   4. D$8$
   5. E$9$

   CEMC Fermat 2025

   Combinatorics Solution and answer checking →

7. A public holiday is always celebrated on the third Wednesday of a
   certain month. In that month, the holiday cannot occur on which of the
   following days?

   1. A16th
   2. B22nd
   3. C18th
   4. D19th
   5. E21st

   CEMC Gauss (Grade 8) 2022

   Combinatorics Solution and answer checking →

8. 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. AAbel and Callie
   2. BBrock and Duan
   3. CAbel and Brock
   4. DBrock and Callie
   5. ECallie and Duan

   CEMC Gauss (Grade 8) 2026

   Combinatorics Solution and answer checking →

9. Glen, Hao, Ioana, Julia, Karla, and Levi participated in the 2023
   Canadian Team Mathematics Contest. On their team uniforms, each had a
   different number chosen from the list $11$, $12$, $13$, $14$, $15$, $16$. Hao’s and Julia’s numbers were even.
   Karla’s and Levi’s numbers were prime numbers. Glen’s number was a
   perfect square. What was Ioana’s number?

   1. A$11$
   2. B$15$
   3. C$14$
   4. D$13$
   5. E$12$

   CEMC Pascal 2024

   Combinatorics Solution and answer checking →

10. In a football tournament participated $n$ teams. Each team played exactly one match with each other team. There was a total of $2015n$ matches played in the tournament. How many teams participated in the tournament?

    1. A2015
    2. B4029
    3. C4030
    4. D4031
    5. EIt is impossible to determine

    National Math Olympiad – First Round 2015

    Combinatorics Solution and answer checking →

Answer key — Stage 2 · Combinatorics

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

1. $B$ open
2. $Bob$ open
3. $D$ open
4. $B$ open
5. $A$ open
6. $A$ open
7. $D$ open
8. $A$ open
9. $A$ open
10. $D$ open

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