, 2022

a) Obviously $k=1$ fails (e.g., whenever a couple of neighbors share their favorite number). We will show that $k=2$ works. If all of them share the same favorite numbers, say 1 and 2, then we can easily solve the problem via providing 7 T-shirts of each kind, since 14 is even. If this is not the case, WLOG we can order the participants, such that the first one has a favorite number $A$, which is not among the favorites of the last one. Then we give an $A$ T-shirt to the first. For each consecutive participant, we choose a favorite number, different than the one ahead of him. This will always be possible, thus the answer is $k=2$.

b) First, $k \le 3$ fails when the favorite numbers of both boys and girls form the triples $\{1,2,3\}$, $\{1,4,5\}$, $\{1,6,7\}$, $\{2,4,6\}$, $\{2,5,7\}$, $\{3,4,7\}$, $\{3,5,6\}$. We will show that $k=4$ works. It is enough to consider exactly 4 favorite numbers for each of the boys, and all the different ways to assign a symbol $B$ or $G$ to each of the numbered T-shirts, for which there are no T-shirts with different symbols but identical numbers. Such a labeling will be referred as "good". Note that, there are at least $2^4 = 16$ "good" labelings (when all the girls and all the boys share the same set of favorite numbers), and the probability for a boy to have all his favorite T-shirts labeled $G$ is exactly $\frac{1}{16}$ from all good cases. Analogously, for a girl to have $B$ on all her favorite T-shirts. Therefore, in at most $14 \cdot \frac{1}{16} = \frac{7}{8}$ of the "good" cases the organizers will fail to distribute the T-shirts; hence in at least $\frac{1}{8}$ of the good cases (in particular, in at least two) the organizers can accomplish their goal.