Find the mistake in the below maths equations

A = 2
A(A-1) = 2(A-1)
A2-A = 2A-2
A2-2A = A-2
A(A-2) = A-2
A = 1

What is the largest amount of money that you can possess in change and still if someone asks you for a dollar, you won't be able to give?

I am a seven-letter word but if you remove four of my letters I will have one left. What word am I?

Can you find the thief by just looking at the picture below?

A wooden log balances equally on the scales when weighed with three-quarters of a pound and three-quarters of a wooden log.

Can you calculate the weight of the entire wooden log?

There are people and strange monkeys on this island, and you can not tell who is who (Edit: until you understand what they said - see below). They speak either only the truth or only lies.
Who are the following two guys?
A: B is a lying monkey. I am human.
B: A is telling the truth.

There are forty elephants and they have forty-fore heads. How can this be possible?

John was piloting a plane behind a car but was never able to overtake it. Why?

A large container is kept in open under the rain. Every passing hour, the water collected inside the container becomes double what it was.

In ten hours, the container is filled completely. Can you calculate how long would it have taken to be filled in half?

It's pretty hard to give up.
If you remove a part of it, you will be left with a bit.
Even if you remove another part, the bit still remains.
Remove one more and it still remains.

What is it?