2, 3, 5, 9, 17, _ What is the next number in the sequence?

I come in different shapes and sizes.
Parts of me are curved, other parts are straight.
You can put me anywhere you like,
but there is only one right place for me.
What am I?

Can you find out the remainder when 3^300 is divided by 5?

A chess tournament is taking place on knock-out terms (the one who loses the match is out of the game).
(a) If 10 matches are played in total, how many players participated?
(b) If 20 players took part in the tournament, how many matches were played?

Here is a mathematical expression that also secretly tells a movie name. What movie is that?

Two guards were on duty outside a barracks. One faced up the road to watch for anyone approaching from the North. The other looked down the road to see if anyone approached from the South. Suddenly one of them said to the other, "Why are you smiling?"

How did he know his companion was smiling?

Rearrange the letters in "new door" to make one word.

The following question it puts forth you:

25 - 55 + (85 + 65) = ?

Then, you are told that even though you might think it's wrong, the correct answer is actually 5!

Whats your reaction to it? How can this be true?

Twice ten are six of us,

Six are but three of us,

Nine are but four of us;

What can we possibly be?

Would you know more of us?

Twelve are but six of us,

Five are but four, do you see?

What are we?

What number comes next in this sequence:
7 8 5 5 3 4 4 _