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.
But try the problem first...
Answer: there is at least one person who shakes hand with an even number of persons
B.Stat Objective Question 63
Challenges and Thrills of Pre-College Mathematics by University Press
Let R be handshakes among 99 persons holds
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
then there exists at least one person who shakes hand with an even number of persons.