Problem A. 923. (January 2026)

A. 923. \(\displaystyle 2026\) people visited an exhibition where \(\displaystyle 1000\) paintings were displayed. Prove that it is possible to send some of the visitors to two rooms, with at least one visitor in each room, such that there is no painting that was liked by someone in one room but by nobody in the other, and there is no painting whose painter is personally known by someone in one room but by nobody in the other.

(7 pont)

Deadline expired on February 10, 2026.

For each visitor, construct a \(\displaystyle 0\)–\(\displaystyle 1\) vector \(\displaystyle \mathbf{v}_j\in\mathbb{R}^{2000}\) as follows. In the first \(\displaystyle 1000\) coordinates, we put a \(\displaystyle 1\) in position \(\displaystyle i\) if the \(\displaystyle i\)th painting was liked by the given visitor, and \(\displaystyle 0\) otherwise. In the second \(\displaystyle 1000\) coordinates, we put a \(\displaystyle 1\) in position \(\displaystyle 1000+i\) if the visitor knows the painter of the \(\displaystyle i\)th painting, and \(\displaystyle 0\) otherwise.

Since we have \(\displaystyle 2026\) vectors in \(\displaystyle \mathbb{R}^{2000}\), they cannot be linearly independent. Thus there exist real numbers \(\displaystyle \lambda_1,\lambda_2,\dots,\lambda_{2026}\), not all zero, such that

\(\displaystyle \lambda_1\mathbf{v}_1+\lambda_2\mathbf{v}_2+\dots+\lambda_{2026}\mathbf{v}_{2026}=0. \)

Let \(\displaystyle I^+=\{i:\lambda_i>0\}\) and \(\displaystyle I^-=\{i:\lambda_i<0\}\). One of these sets is certainly nonempty, and since all coordinates of the vectors are nonnegative, the other one cannot be empty either. We claim that these two sets provide a suitable partition of the visitors. Indeed, if in one group a painting is liked by someone, then in the other group it must also be liked by someone; otherwise the coordinate corresponding to that painting among the first \(\displaystyle 1000\) coordinates could not be zero in the linear combination. Of course, the same argument applies to the second \(\displaystyle 1000\) coordinates as well.

Statistics:

20 students sent a solution.
7 points:	Ali Richárd, Aravin Peter, Bodor Mátyás, Diaconescu Tashi, Gyenes Károly, Kis Ágoston, Rajtik Sándor Barnabás, Sárdinecz Dóra, Szakács Ábel, Vincze Marcell, Xiaoyi Mo.
6 points:	Bolla Donát Andor, Forrai Boldizsár.
2 points:	1 student.
1 point:	4 students.
0 point:	2 students.

Problems in Mathematics of KöMaL, January 2026