# 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

Solution

### 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.

Hints

Solution