May 2026 Contest — Grade 6

Question 1

Coach Marcus has six runners with different speeds on his track team. He needs to select four runners for a relay race where the first leg runner must be slower than the second, who must be slower than the third, who must be slower than the fourth (anchor). Once four runners are chosen, there is only one valid way to arrange them by this rule. How many different relay teams could Coach Marcus form?

Accepted Answer

To find the number of different relay teams Coach Marcus can form, we need to select 4 runners out of the 6 available runners.

The rule states that the first runner must be slower than the second, who must be slower than the third, who must be slower than the fourth. Since all six runners have different speeds, once we choose any group of 4 runners, there is only one unique way to arrange them from slowest to fastest.

Therefore, the number of different relay teams is simply the number of ways to choose 4 runners from a group of 6.

Choosing 4 runners to be on the team is mathematically the same as choosing 2 runners to leave off the team. We can find the number of ways to choose 2 runners to leave out from 6:
If we label the runners A, B, C, D, E, and F, the possible pairs to leave out are:
 1. AB
 2. AC
 3. AD
 4. AE
 5. AF
 6. BC
 7. BD
 8. BE
 9. BF
 10. CD
 11. CE
 12. CF
 13. DE
 14. DF
 15. EF

There are exactly 15 different ways to choose which 2 runners to leave out, which means there are 15 different ways to choose the 4 runners for the relay team.

Thus, Coach Marcus can form 15 different relay teams.

The answer is C

Question 2

A diagram shows direct flight routes between six cities: Athens, Berlin, Cairo, Dublin, Edinburgh, and Florence. Each number (1-6) represents one city, and each line connecting two numbers represents a direct flight between those cities. Cairo has more connections than Edinburgh but less than Dublin. Berlin has direct flights to only Dublin and Athens. Which number represents Cairo?

Accepted Answer

To solve this problem, let's analyze the diagram step-by-step:

Step 1 — Read the image
Let's count the number of direct flight connections (lines) for each numbered city (vertex) in the diagram:
City 1 is connected to: 3, 4 (Total: 2 connections)
City 2 is connected to: 3, 4, 5 (Total: 3 connections)
City 3 is connected to: 1, 2, 4, 5, 6 (Total: 5 connections)
City 4 is connected to: 1, 2, 3, 5, 6 (Total: 5 connections)
City 5 is connected to: 2, 3, 4, 6 (Total: 4 connections)
City 6 is connected to: 3, 4, 5 (Total: 3 connections)

Step 2 — Solve
1. We are given that Berlin has direct flights to only Dublin and Athens. This means Berlin has exactly 2 connections. Looking at our counts, City 1 is the only city with exactly 2 connections. Thus, City 1 represents Berlin.
2. Since Berlin (City 1) is connected to Dublin and Athens, these two cities must be represented by City 3 and City 4 (in some order). Both of these cities have exactly 5 connections. Therefore, Dublin has 5 connections.
3. We are told that Cairo has more connections than Edinburgh but less than Dublin. Since Dublin has 5 connections, Cairo must have fewer than 5 connections.
4. The remaining cities (Cairo, Edinburgh, and Florence) must be represented by the remaining numbers: 2, 5, and 6, which have 3, 4, and 3 connections respectively.
5. For Cairo to have more connections than Edinburgh but fewer than Dublin (5), Cairo must have exactly 4 connections, and Edinburgh must have 3 connections (since 3 < 4 < 5).
6. The only city with exactly 4 connections is City 5.

Therefore, City 5 represents Cairo.

Step 3 — Select
This corresponds to option D.

The answer is D

May 2026 Contest — Grade 6

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