Solution:

58 minutes

If you see the net progress of the hiker, it is one feet per minute. The most common answer is therefore 60 minutes or an hour. But if you consider the situation, after 58 minutes, he would have climbed the mountain and would be on top of it, thus he will not slip back.

Solution:

Keep one coin from the first set and place it on the scale along with the two from the second set etc. If the weight is off by one hundredth of an ounce, then you will know that it is the first set that is faulty and if the weight is off by two hundred of an ounce, then the second set is faulty and so on.

Solution:

This question is one of the most common interview question. The most common answer of the question is sixty which is completely wrong. The second most common answer of this interview question is 10 which is wrong as well.

What we have is
6 fishermen catches 6 fishes in 6 minutes

If you offer them 10 times as much as time, they will surely catch 10 times as much as fishes as well
Thus the same six fishermen will be able to catch 60 fishes in 60 minutes.

You can also thing this in the unitary method.

If six fishermen catches 6 fishes in 6 minutes
Then, one fisherman will catch one 1 fish in 6 minutes
Thus in sixty minutes, one fisherman will catch 10 fishes (10 times the time, ten times the fishes).

Thus the right answer is six.

Solution:

He can ask any one of them. The question he will ask is - "Which door will the other guy say is the correct one?"

If he asks this question to the truth teller, he will get the incorrect door as he will honestly tell him what the liar will say. If he asks the liar, he will get the incorrect door again as he will lie about what the truth teller will tell him (the truth teller will obviously tell him the right door). Thus, by asking any of them, Sheldon Cooper will get the incorrect door only and he can choose the other one to find the treasure.

Solution:

First time = 20 ft
Second time = 10ft down + 10 ft up
Third time = 5ft up + 5ft down
It will go on like this the object stops

Total distance traveled = 20 + (10 + 10) + (5 + 5) + (2.5 + 2.5) + . . .
= 20 + (10 + 5 + 2.5 + ...) + (10 + 5 + 2.5 + ...)
Assuming the object comes to rest at infinity,

Total distance traveled = 20 + [10/ (1 - 0.5)] + [10/ (1 - 0.5)]
= 20 + 20 + 20 = 60

Thus option d is correct.