Problem A. 899. (February 2025)

A. 899. The world famous infinite hotel with infinitely many floors (where the floors and the rooms on each floor are numbered with the positive integers) is full of guests: each room is occupied by exactly one guest. The manager of the hotel wants to carpet the corridor on each floor, and an infinite set of carpets of finite length (numbered with the positive integers) was obtained. Every guest marked an infinite number of carpets that they liked. Luckily, any two guests living on a different floor share only a finite number of carpets that they both like. Prove that the carpets can be distributed among the floors in a way that for every guest there are only finitely many carpets they like that are placed on floors different from the one where the guest is.

Proposed by: András Imolay, Budapest

(7 pont)

Deadline expired on March 10, 2025.

Let's enumerate the guests (this is possible because of the well known fact that the pairs of positive integers form a countable set). Now let's distribute the carpets according to the following rule: if the set of guests liking the given carpet is non-empty, then find the guest that is the first in the enumeration, and place the carpet on their floor. Those carpets that are not liked by any of the guests can be placed anywhere, e.g. all can be placed on the first floor.

Let's prove that this distribution works: let's consider the guest who is the \(\displaystyle n^\text{th}\) in the enumeration. Now those carpets that they like but are not placed on their floor have to be liked by some other guest who comes earlier in the enumeration. However, the number of such guests is finite, and each of them likes only finitely many of the carpets that are liked by the \(\displaystyle n^\text{th}\) guest, therefore their total number is still finite, which proves our claim.

Statistics:

18 students sent a solution.
7 points:	Aravin Peter, Balla Ignác , Bodor Mátyás, Czanik Pál, Forrai Boldizsár, Holló Martin, Keresztély Zsófia, Kocsis 827 Péter, Morvai Várkony Albert, Sánta Gergely Péter, Szakács Ábel, Szaszkó Benedek, Varga Boldizsár, Virág Tóbiás, Vödrös Dániel László, Xiaoyi Mo.
0 point:	1 student.
Not shown because of missing birth date or parental permission:	1 solutions.

Problems in Mathematics of KöMaL, February 2025