MathDB

For a competition a school wants to nominate a team of

k

students, where

k

is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the

n \ge 2

students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition:

(*)

If a student

X

is not nominated for the team, there is a student

Y

on the team who defeated

X

in at least two disciplines.
Determine all positive integers

n \ge 2

such that for any combination of rankings, a team can be chosen to satisfy the condition

(*)

, when
a)

k=2

, b)

k=3

Problem Statement