MathDB

5

Part of 2022 Durer Math Competition (First Round)

Problems(2)

game master divides a group of 12 players into 2 teams of 6

Source: (2021 -) 2022 Dürer Math Competition Regional E5 https://artofproblemsolving.com/community/c1621671_

11/30/2022
a) A game master divides a group of 1212 players into two teams of six. The players do not know what the teams are, however the master gives each player a card containing the names of two other players: one of them is a teammate and the other is not, but the master does not tell the player which is which. Can the master write the names on the cards in such a way that the players can determine the teams? (All of the players can work together to do so.)
b) On the next occasion, the game master writes the names of 33 teammates and 11 opposing player on each card (possibly in a mixed up order). Now he wants to write the names in such away that the players together cannot determine the two teams. Is it possible for him to achieve this?
c) Can he write the names in such a way that the players together cannot determine the two teams, if now each card contains the names of 44 teammates and 11 opposing player (possibly in a mixed up order)?
combinatorics
a_i = -a_{n+1-i} if sum a_i^{2k+1} = 0 when a_1 <= a_2 <= ... <= a_n

Source: (2021 -) 2022 Dürer Math Competition Regional E+5 https://artofproblemsolving.com/community/c1621671_

11/30/2022
Let a1≤a2≤...≤ana_1 \le a_2 \le ... \le a_n be real numbers for which ∑i=1nai2k+1=0\sum_{i=1}^{n} a_i^{2k+1} = 0 holds for all integers 0≤k<n0 \le k < n. Show that in this case, ai=−an+1−ia_i = -a_{n+1-i} holds for all 1≤i≤n1 \le i \le n.
algebraSequence