Solution:

Seven

Dividing the horses in a group of five, we will have five different groups of horses. A race will be conducted separately for each group which will help us find the fastest three horses in every group. We have used five different races till now.

Now we can have a race among the winners from every race which will help us determine the fastest of them all. We have used the sixth race.

The second fastest horse can be the one who came second in the sixth race or the second fastest horse in the finals winners' first race.

If we look at the first case, the third fastest horse can be the third fastest horse in the sixth race or the second fastest horse in either the fastest or second fastest horse first race.
If we look at the other case, the third fastest horse may be the second fastest in the sixth race or the third fastest in the final winners' first race.

If you calculate our assumptions, we have five more horses to race (seventh race) and determine the second fastest and the third fastest horse.

Solution:

nineteen

Explanation:
worstcase (7purple+11yellow+1blue)

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:

You can do it by two methods. However, we are explaining only one and leaving one for you to figure out yourself.

What you will do is fill the 3 liters jar and pour it into the 5 liters jar. Fill the 3 liters jar again and pour it into the 5 liters jar till it's full. Now we know that the 5 liters jar was already having 3 liters of water, thus you would be able to fill 2 liters only. In that case, the 3 liters jar will be left with 1 liter water. Empty the 5 liters jar and then pour the remaining 1 liter from 3 liters jar to the 5 liters jar. Refill the 3 liters jar and pour it completely into the 5 liters jar which already had 1 liter water. Thus now the 5 liters jar have 4 liters of water.

Solution:

24 times is the most common answer however that is completely wrong. The right answer will be 22 times.

Let us prove it by some simple mathematics.
Assume that it takes T hours for the minute hand to complete T laps. The hour hand will complete T/12 Laps in the same time.

Consider the situation when the hour hand and the minute hand will overlap for the first time; the minute hand would have completed one lap extra than the hour hands.
Thus, T =T/12 + 1

Considering the above expression we know that the first overlap will take place after t = 12/11 hours i.e. 1:05 am.

Similarly, the second overlap will take place when the minute hand would have completed two more laps than the hour hand.

Considering there are X laps,
T = T/12 + X

Everyone knows that a day comprises of 24 hours. Putting that in the equation we get
24 = 24/12 + X
Solve it and you will get
X = 22

Hence both of the hands will overlap 22 times in 24 hours.

Let us give you with those exact timings as well.
The hands will overlap at 12:00, 1:05, 2:10, 3:15, 4:20, 5:25, 6:30, 7:35, 8:40, 9:45, and 10:50. Do consider the fact that there will be no 11:55. It becomes 12:00.