What seven-letter word has hundreds of letters in it?

Can you solve below mathematical equation?

2^1234 - 2^1233

I am under you when I am whole.
I shift above you if you remove the first letter.
I come all around you if you remove the second as well.

Can you tell me who I am?

I look at you, you look at me, I raise my right, you raise your left. What am I?

I am a word of 12 alphabets.

* Alphabets 12, 4, 7, 2, 5 : Eastern beast of burden
* Alphabets 1, 8, 10, 9 are street made famous by Sinclair Lewis.
* Alphabets 11, 3, 6 are past.

On whole it means a person suffering from delusions of greatness.

What Is it ?

One is to three as three is to five and five is to four and four is the magic number.
What is the pattern?

Five lovely ladies(Sophia, Isabella, Madison, Emma and Olivia) planned a picnic.

They each buy one thing each for the picnic.

Sophia, Emma and Olivia got a drink : orange-Juice, apple-juice, and mango-juice.
Olivia got the drink with the same letter as the one in her first name.
Emma loves mango-juice.
The other two bring some food : chocolates and pizza.
Also Madison is allergic to cheese.

So which ladies bought what ?

A girl was sitting in her hotel room when she heard a knock on the door. She opened the door and found that a man was standing outside. The man said, "Oh! I am really sorry, I thought this was my room." He then walked through the corridor to the elevator. The girl did not know the man. She closed her door and called security asking them to apprehend the man. What made her suspicious of that man? He might have been genuinely mistaken.

This is a classy optical illusion puzzle. What more you can find except bricks in the image below?

In the given figure, move three matchsticks so that the resulting figure contains two rectangles.