Solution:

John starts after 3 days, thus Peter has already covered 60 km by then (travelling at 20km a day)
The distance between Peter and John will be increased to 60 + 5 + 1 = 66 km in the next two days.

Since John is following an arithmetic progression in the speed daily, the distance between him and peter will keep on decreasing from the 6th day. It will decrease by 3 km on the first day, 7 km on the next day and so on.

Now John will catch up with Peter after n + 5 days from the start of Peter
Thus
66 = (n/2)*(2*3 + (n-1)*4)

If we solve for n in the above equation, we will get n = 5.5

Thus John will catch up with Peter on the eights day when he starts running after Peter.

Solution:

1/5

There are six possible pairings of the two balls withdrawn,
Yellow+Yellow
Yellow+Green
Green+Yellow
Yellow+Black
Black+Yellow
Green+Black.

We know the Green + Black combination has not been drawn.

This leaves five possible combinations remaining. Therefore the chances tbowl the Yellow + Yellow pairing has been drawn are 1 in 5.

Many people cannot accept tbowl the solution is not 1 in 3, and of course it would be, if the balls had been drawn out separately and the color of the first ball announced as Yellow before the second had been drawn out. However, as both balls had been drawn together, and then the color of one of the balls announced, then the above solution, 1 in 5, must be the correct one.

Solution:

Light both the ends of the first string and one end of the second string. 30 minutes will have passed when the first string is fully burned, which means 30 minutes have burned off the second string. Light the end of the second string and when it is fully burned, 45 minutes will have passed.

Solution:

The nail would be at the same height since trees grow at their tops.

Solution:

Just 1