Math Problems

Here's a small collection of mathematics problems; each one has some logical fallacy in it. See if you can solve the problem before looking at the answer! (Problems 4 and 5 are my favorites).

Problem 1 - A hotel manager fits thirteen people into twelve rooms, each with their own room.
Problem 2 - One dollar is worth 10,000 cents.
Problem 3 - Two times two is five.
Problem 4 - Selling apples doesn't make as much as expected.
Problem 5 - Dividing the inheritance.
Problem 6 - Sum of the infinite series 1 - 1 + 1 ...
Problem 7 - Proof that cos(pi/4) = 1

These were all taken from the book "Lapses in Mathematical Reasoning", by V. M. Bradis, et. al., Dover Publications.