Question 122
At an IMO prep camp there are 14 participants. Each participant has at least $k$ favorite numbers. The organizers want to give to each student a T-shirt with one of their favorite numbers on it. Find the smallest $k$ for which this is always plausible under the restriction:
a) the participants form a circle and the T-shirts must then be distributed in such a way that no two neighbors share the same number.
b) There are 7 boys and 7 girls and neither of the boys share their T-shirt number with neither of the girls.
a) the participants form a circle and the T-shirts must then be distributed in such a way that no two neighbors share the same number.
b) There are 7 boys and 7 girls and neither of the boys share their T-shirt number with neither of the girls.