7

Part of 2022 Saint Petersburg Mathematical Olympiad

Problems(2)

Good sets of cards

Source: St Petersburg 2022 11.7

Given is a set of

2n

1,2, \cdots, n

, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify

80

n

cards, of which at least one is sure to be good. What is the maximal

n

for which the Baron's words could be true?

combinatoricsBaron Munchausen

Permutations of digits and divisibilities

Source: St Petersburg 2022 9.7

n

distinct natural numbers. For any two of them, the one is obtained from the other by permuting its digits (zero cannot be put in the first place). Find the largest

n

such that it is possible all these numbers to be divisible by the smallest of them?