Contradiction study problems

Problem 1

Use a proof by contradiction to show that \(\log_3 2\) is irrational.

Hints for getting started.

Started the proof but got stuck? More Hints


Problem 2

There were 103 people at a bottle cap trader's convention. Every trade involves two people. Show that at least one of these must be true: (1) some trades are profitable for only one person; (2) at least two people made profitable trades with the same number of other people.

Note that someone could make 0 trades, so a person could trade with between 0 and 102 other people.

Also notice that the two people involved in a trade might have different goals. Person A might like green bottle caps and person B might like ones with dragons on them. This is how a trade might benefit both people, or neither person.