MathDB

1

Part of 2013 Argentina National Olympiad

Problems(1)

2013 numberd from 1-2013

Source: 2013 Argentina OMA Finals L3 p1

1/15/2023
On a table there are 20132013 cards that have written, each one, a different integer number, from 11 to 20132013; all the cards face down (you can't see what number they are). It is allowed to select any set of cards and ask if the average of the numbers written on those cards is integer. The answer will be true. a) Find all the numbers that can be determined with certainty by several of these questions. b) We want to divide the cards into groups such that the content of each group is known even though the individual value of each card in the group is not known. (For example, finding a group of 33 cards that contains 1,21, 2, and 33, without knowing what number each card has.) What is the maximum number of groups that can be obtained?
combinatorics