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Try this TOMATO problem from I.S.I. B.Stat Objective based on Relations and Numbers.

We consider the relation , "a person x shakes hand with a person y".Obviously if x shakes hand with y, then y shakes hand with x. In a gathering of 99 persons , one of the following statements is always true, considering 0 to be an even number, find which one is it.

- there is at least one person who shakes hand with an odd number of persons
- there is at least one person who shakes hand with an even number of persons
- there are even number of persons who shake hand exactly with an even number of persons
- none of these

Logic

Relations

Numbers

But try the problem first...

Answer: there is at least one person who shakes hand with an even number of persons

Source

Suggested Reading

B.Stat Objective Question 63

Challenges and Thrills of Pre-College Mathematics by University Press

First hint

Let R be handshakes among 99 persons holds

Second Hint

first person may handshake with at most 98(even) other persons, for second person similar arguments hold and this holds with similar arguments for all persons

Final Step

then there exists at least one person who shakes hand with an even number of persons.

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