Opgaver til kursusgang 1.

1.1.44
An ancient Sicilian legend says that the barber in a remote town who can be reached only by travelling a dangerous mountain road shaves those people, and only those people who do not shave themselves. Can there be such a barber.

1.6.7
Use a direct proof to show that every odd integer is the difference of two squares.

1.6.23
Show that at least 10 of any 64 days chosen must fall on the same day of the week.

1.6.39
Prove that at least one of the real numbers a₁, a₂, ... , a_n is greater that or equal to the average of these numbers.
What kind of proof did you use?

1.6.40
Use Exercise 39 to show that if the first 10 positive integers are placed around a circle, in any order, there exist three integers in consecutive locations around the circle that have a sum greater than or equal to 17.


Opgave til afsnit 1.6: Vis at log₂(3) er irrational.
(19/12 bliver dog anvendt som en approximation. Se eventuelt appendix A-2 om logaritme funktioner).