What has to be broken before you can use it?

A boy and a girl are sitting on the porch.
"I'm a boy," says the child with black hair.
"I'm a girl," says the child with red hair.
If at least one of them is lying, who is which?

Solve the following series:
AZ, GT, MN, __, YB

You have two buckets - one holds exactly 5 gallons and the other 3 gallons. How can you measure 4 gallons of water into the 5 gallon bucket?

(Assume you have an unlimited supply of water and that there are no measurement markings of any kind on the buckets.)

What has hands, but can’t clap?

What's the best place to go shopping in the ocean?

How many bananas can you eat if your stomach is empty?

A farmer and his neighbour once went to Emperor Akbar"s court with a complaint. "Your Majesty, I bought a well from him," said the farmer pointing to his neighbour, "and now he wants me to pay for the water."
"That"s right, your Majesty," said the neighbour. "I sold him the well but not the water!"?
The Emperor asked Birbal to settle the dispute. How did Birbal solve the dispute?

There are nine letters in the below-given diagram. You have to form a nine-letter word using these letters.

You see a mathematical expression in the picture. Can you think of a movie that this expression refers to?