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 .            

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Similar Problem

The Mathematics of How Virus can Grow

The Mathematics of How Corona Virus Grow? The beautiful tale of undeterministic mathematics of chance and chaos of when they will become extinct or when they will thrive.

The Exaggerated Triangle Inequality

Triangle Inequality is an exaggerated version of the Basic Idea of the Euclidean Plane. Let’s do some Triangle inequality Problems and Solutions.

Geometric Median |Understand the concept

Geometric Median is an important concept in the intersection of Geometry, Data Analysis and Algorithms. This article explores the concept.

Examples & Counterexamples – A Way to Build Your Own Mathematics

This is an interesting article on how to build your own Mathematics with the help of examples and counter examples. Stay tuned.

Sets and Venn diagrams |B.Math Entrance

Try this beautiful problem from B.Math Entrance Exam based on sets and venn diagrams. You may use sequential hints to solve the problem.

INMO 2007

Try to solve these interesting INMO 2007 Questions. Solve them and write the answers in the comment to check your answers.

Order of General and Special Linear Group

Here is the post in which you would learn about the Order of General and Special Linear Group with the help of a problem. Try it and learn the solution.

Maximizing Arrangements

Here is a post related to a problem based on maximizing arrangements in Mathematics. Try the problem and learn the solution.

Gaps in Permutation | TOMATO Objective Problem

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

Geometry of Tangents | ISI Entrance B.Stat 2009

Objective Problem Geometry (ISI Entrance) Find the radius of smaller circle. 01$\frac{3}{4}$2 Key Concepts 2D Geometry Similar Triangles Linear Equations Check the Answer Answer: $\frac{3}{4}$ ISI Entrance B.Stat Objective Problem, India Test of Mathematics at 10+2...