Understand the problem

In a badminton singles tournament, each player played against all the others
exactly once and each game had a winner. After all the games, each player
listed the names of all the players she defeated as well as the names of all the
players defeated by the players defeated by her. For instance, if A defeats B
and B defeats C, then in the list of A both B and C are included. Prove that
at least one player listed the names of all other players.

Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2013. Subjective Problem no. 4.

Topic
combinatorics

Difficulty Level

7.5 out of 10

Suggested Book

Problem Solving Strategies by Engel

 

Start with hints

Do you really need a hint? Try it first!

Do you know what is Well-ordering principle ? it says that Every nonempty set A of nonnegative integers has a minimal element and a maximal element , which need not to be unique . now some time this simple property help us to get very nice solution to a problem , can you some how apply this property .      

Use the method of contradiction , first of all assume that there is no player which have the given property . Now try to use the property of hint 1 .  

If there is no such list , A’s list has the maximum no. of players Now , if A does not have the certain property then there exist another another player B , who has won against A . Now B’s list contain the name of A [ by the 1st condition ] and all the names of the players defeated by A [ by the 2nd condition]  

Now , can you find out some contradiction , yes exactly ….. B’s list contain more number of element than A So, A’s list must have the certain property .      

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