MathDB

4

Part of 2023 Middle European Mathematical Olympiad

Problems(2)

(n, m)-good sets

Source: MEMO 2023 I4

8/24/2023
Let n,mn, m be positive integers. A set SS of positive integers is called (n,m)(n, m)-good, if:
(1) m∈Sm \in S; (2) for all a∈Sa\in S, all divisors of aa are also in SS; (3) for all distinct a,b∈Sa, b \in S, an+bn∈Sa^n+b^n \in S.
For which (n,m)(n, m), the only (n,m)(n, m)-good set is N\mathbb{N}?
number theoryMEMO2023
Teams with non-clashing uniforms

Source: MEMO 2023 T4

8/25/2023
Let c≥4c \geq 4 be an even integer. In some football league, each team has a home uniform and anaway uniform. Every home uniform is coloured in two different colours, and every away uniformis coloured in one colour. A team’s away uniform cannot be coloured in one of the colours fromthe home uniform. There are at most cc distinct colours on all of the uniforms. If two teams havethe same two colours on their home uniforms, then they have different colours on their away uniforms. We say a pair of uniforms is clashing if some colour appears on both of them. Suppose that for every team XX in the league, there is no team YY in the league such that the home uniform of XX is clashing with both uniforms of YY. Determine the maximum possible number of teams in the league.
combinatorics