What word is represented by this rebus?

I have no doors but I have keys, I have no rooms but I have space, you can enter but you can’t leave! What am I?

A beggar on the street can make one cigarette out of every 6 cigarette butts he finds. After one whole day of searching and checking public ashtrays the beggar finds a total of 72 cigarette butts. How many cigarettes can he make and smoke from the butts he found?

Why is 6 afraid of 7?

A convention is held where all the big logicians are summoned. The master places a band on everyone's forehead. Now all of them can see others bands but can't see his own. Then they are told that there are different colours of bands. All the logicians sit in circle and they are further explained that a bell will ring at regular intervals. The moment when a logician knew the colour of band on his forehead, he will leave at the next bell. If anyone leaves at the wrong bell, he will be disqualified.

The master assures the logicians that the puzzle will not be impossible for anyone of them. How will the logicians manage ?

How many points are there on the globe where, by walking one mile south, then one mile east and then one mile north, you would reach the place where you started?

If,

6 / 7 = 2
3 / 9 = 7
8 / 2 = 6

Then

4 / 5 = ?

Can you find the key in the picture below?

Replace the question mark with the correct number.

Only one color, but not one size,
Stuck at the bottom, yet easily flies.
Present in sun, but not in rain,
Doing no harm, and feeling no pain.
What is it?