Solution:

The total population of New York City is about nine millions. Let us assume that 1 million among them are bald which leaves us with eight millions of people that are required to have different number of hairs on their head.

Now as per the surveys, on an average, people have 100, 000 hairs on their head. Since all must have different numbers, we start with one. Let us say someone has one hair strand, someone has two, and someone has three and so on. After 100, 000 people, we are left with 7, 900, 000 people left. Since all of them are needed to have different number of hair strands, we will move up from 100, 000.

In such case, at the end, you will be having a person with 8, 000, 000 hairs which is clearly impossible (even having 1, 000, 000 hairs on someone's head is impossible).

Therefore, there must be two people who have same number of hairs on their head.

This is a classic example of the pigeonhole principle.

Solution:

At first, there were 2 apples on the tree. After the wind blew, one apple fell on the ground. So there where no apples on the tree and there were no apples on the ground.
*apples => more than one apple

Another solution(non tricky one)
The wind blew so hard that the apples fell of the tree and blew along the ground.

Solution:

Since the door is not visible in this picture, the bus is moving in the left direction.

Solution:

Yes, some people are going get really tiny pieces but you were allowed for that right. Also, there is another way of doing it. Can you think of it?

Solution:

In the picture you can find the green matchsticks. We have moved them from the bottom and placed them in the middle. It was a bit tricky but now you have your desired result.

Solution:

Just 2

Day One: You make your first cut at the 1/7th mark and give that to the worker.
Day Two: You cut 2/7ths and pay that to the worker and receive the original 1/7th in change.
Day three: You give the worker the 1/7th you received as change on the previous day.
Day four: You give the worker 4/7ths and he returns his 1/7th cut and his 2/7th cut as change.
Day Five: You give the worker back the 1/7th cut of gold.
Day Six: You give the worker the 2/7th cut and receive the 1/7th cut back in change.
Day Seven: You pay the worker his final 1/7th.

Solution:

This puzzle has no answer if you are considering this figure in a 2d domain. However if you follow it on a three-dimensional plane, you can do it like shown in the figure given.

Solution:

The guard and the prisoner cross the river
The guard comes back
The guard and the girl cross the river.
The guard and the prisoner comes back.
The mother and the girl cross the river.
Mother comes back
The mother and the father cross the river
Father comes back
The guard and the prisoner cross the river
Mother comes back
The father and the mother cross the river
Father comes back
The father and the boy cross the river
The guard and the prisoner comes back
The guard and the boy cross the river
The guard comes back
The guard and the prisoner cross the river.

Solution:

The brain teaser can be solved as shown in the image below.

Solution:

Adam and Eve were the only people without navel. As they were not born out of a mother, they never had umbilical cords and thus never had navels.

Solution:

Move and then pour all juice from second glass to fifth glass.