What is the Surname of Barbie Doll?

In a boat, the father of a sailor's son is sitting with the son of the sailor. However, the sailor is not present on the boat.

Can this even be possible?

When you say goodbye, I am what you need to tie.
I come as a couple and go wherever you run.
Who am I?

There is an exact a week gap between Christmas and New Year. Hence, It is obvious that the new year that comes right after Christmas comes on the same day of the week.

A Strange thing happened in the Year 1777. Christmas occurs on Wednesday and New Year on Monday. How is that possible?

Take away my first letter, then take away my second letter. Then take away the rest of my letters, yet I remain the same. What am I?

A journalist was investigating a sacred cult in the jungles of Africa when he was caught by one of the person. He was confined in a cave till the leader arrived. The leader told him that he need to tell him a statement. If he thinks the statement is true, then the journalist head will be chopped off. If he thinks that the statement is false, his head will be smashed with a hammer.

What statement did the journalist make to survive?

There is a box in which distinct numbered balls have been kept. You have to pick two balls randomly from the lot.

If someone is offering you a 2 to 1 odds that the numbers will be relatively prime, for example
If the balls you picked had the numbers 6 and 13, you lose $1.
If the balls you picked had the numbers 5 and 25, you win $2.

Will you accept that bet?

I am a five-letter word under you,
Remove the first letter and I am above you,
Remove the second as well and I am around you.

Who am I?

Can you count the number of squares in the given picture?

Below, you will find the mathematical proof that 10 equals 9.99999?. But is that possible or there is something wrong about it? Can you find the error?

x = 9.999999...
10x = 99.999999...
10x - x = 90
9x = 90
x = 10