If you're 8 feet away from a door and with each move you advance half the distance to the door. How many moves will it take to reach the door?

You will never reach the door! If you only move half the distance, then you will always have half the distance remaining no matter, how small is the number.

Betty signals to the headwaiter in a restaurant, and says, "There is a fly in my tea."
The waiter says "No problem Madam. I will bring you a fresh cup of tea."
A few minutes later Betty shouts, "Get me the manager! This is the same cup of tea."
How did she know?
Hint: The tea is still hot.

Betty had already put sugar in her tea before sending it back. When the "new" cup came, it was already tasted sweet.

At a dinner party, many of the guests exchange greetings by shaking hands with each other while they wait for the host to finish cooking.
After all this handshaking, the host, who didn't take part in or see any of the handshaking, gets everybody's attention and says: "I know for a fact that at least two people at this party shook the same number of other people's hands."
How could the host know this? Note that nobody shakes his or her own hand.

Assume there are N people at the party.
Note that the least number of people that someone could shake hands with is 0, and the most someone could shake hands with is N-1 (which would mean that they shook hands with every other person).
Now, if everyone at the party really were to have shaken hands with a different number of people, then that means somone must have shaken hands with 0 people, someone must have shaken hands with 1 person, and so on, all the way up to someone who must have shaken hands with N-1 people. This is the only possible scenario, since there are N people at the party and N different numbers of possible people to shake hands with (all the numbers between 0 and N-1 inclusive).
But this situation isn't possible, because there can't be both a person who shook hands with 0 people (call him Person 0) and a person who shook hands with N-1 people (call him Person N-1). This is because Person 0 shook hands with nobody (and thus didn't shake hands with Person N-1), but Person N-1 shook hands with everybody (and thus did shake hands with Person 0). This is clearly a contradiction, and thus two of the people at the party must have shaken hands with the same number of people.
Pretend there were only 2 guests at the party. Then try 3, and 4, and so on. This should help you think about the problem.
Search: Pigeonhole principle

After recent events, Question Mark is annoyed with his brother, Skid Mark. Skid thought it would be funny to hide Question's wallet. He told Question that he would get it back if he finds it. So, first off, Skid laid five colored keys in a row. One of them is a key to a room where Skid is hiding Question's wallet. Using the clues, can you determine the order of the keys and which is the right key?
Red: This key is somewhere to the left of the key to the door.
Blue: This key is not at one of the ends.
Green: This key is three spaces away from the key to the door (2 between).
Yellow: This key is next to the key to the door.
Orange: This key is in the middle.

The order (from left to right) is Green, Red, Orange, Blue, Yellow. The blue key is the key to the door.

A grandfather's clock chimes the appropriate number of times to indicate the hour, as well as chiming once at each quarter hour. If you were in another room and hear the clock chime just once, what would be the longest period of time you would have to wait in order to be certain of the correct time?

You would have to wait 90 minutes between 12:15 and 1:45. Once you had heard seven single chimes, you would know that the next chime would be two chimes for 2 o'clock.

If a farmer has 5 haystacks in one field and 4 haystacks in the other field, how many haystacks would he have if he combined them all in the center field?

One. If he combines all of his haystacks, they all become one big stack.