Maths Olympiad Prep

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Combinatorics Difficulty 6.7 National olympiad Find the answer

On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.

A number or a short expression. Spacing and $ signs are ignored.

Solution

To determine whether a line-up consisting of 6 universal candidates is appropriate, we need to carefully analyze the definitions provided in the problem:

1. Appropriate Line-up: This is a selection of 6 players out of 100 candidates that the coach deems capable of forming the team.

2. Perspective Set: A set of 5 candidates is considered perspective if adding one more candidate results in an appropriate line-up.

3. Universal Candidate: A player is universal if they can complete every perspective set of 5 candidates (that does not include them) to form an appropriate line-up.

Given these definitions, let's explore the reasoning:

- Suppose we have a line-up composed exclusively of 6 universal candidates, denoted by C1,C2,,C6C_1, C_2, \ldots, C_6.

- Consider any subset of 5 candidates from this line-up, say {C1,C2,C3,C4,C5} \{C_1, C_2, C_3, C_4, C_5\} . Because all these candidates are universal, adding the sixth candidate C6 C_6 to this subset must yield an appropriate line-up. This follows directly from the definition of a universal candidate, which can complete any perspective set into an appropriate line-up.

- This reasoning holds for any choice of the 5 candidates from the 6 universal candidates, repeatedly reinforcing that adding the remaining (sixth) universal candidate results in an appropriate line-up.

Therefore, the configuration of the 6 universal candidates inherently satisfies the conditions for forming an appropriate line-up:

Yes \boxed{\text{Yes}}

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.