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Steiner Club

 Published on Sunday, 11th October 2026, 05:00 am and solved by 25

Problem 1014

A club has $7$ members, named $\mathtt A$ to $\mathtt G$. Every night, $3$ of the $7$ members come to the club. After several nights, they find that for every combination of $2$ members, there is exactly one night when they both come.

If the members who come on the first $2$ nights are $\mathtt{ABC}$ and $\mathtt{BDF}$ respectively, then on the night that $\mathtt E$ and $\mathtt G$ both come, the members who come must be $\mathtt{BEG}$.

A larger club has $24$ members, named $\mathtt A$ to $\mathtt X$. Every night, $8$ of the $24$ members come to the club. After several nights, they find that for every combination of $5$ members, there is exactly one night when they all come.

Here are the members who come on the first $11$ nights:

ABCDKMPR
ABJKLMNW
ACIOQTUW
ADFMPQTU
AEFNPTVX
AEFQRSTW
AEGIMQSU
BDEKORUX
BEFGHIKU
DEFHKOSW
FHIJKRST

Find the members who come on the night that $\mathtt{CGJLV}$ all come. Enter your answer by concatenating the names in alphabetic order.



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