Solution:

2418

Explanation:
The rule that is being followed here is

P ~ Q ~ R ~ S = [first digit of (P * S)] [last digit of (Q * R)] [first digit of (Q * R)] [last digit of (P * S)]

Using the same rule in the incomplete equation,
7 ~ 2 ~ 7 ~ 4 = [first digit of (7 * 4)] [last digit of (2 * 7)] [first digit of (2 * 7)] [last digit of (7 * 4)]
= 2418

Thus 2418 is the missing number.

Solution:

Yes, this can be possible if the match was between two female boxers.

Solution:

The missing letter on left is P and the missing letter on the right is C. The word starts from the missing letter C and forms the word champion.

Solution:

4

Explanation:
5 = Five (four letters) => 4
7 = Seven (five letters) => 5
10 = Ten (three letters) => 3
3 = Three (five letters) => 5

Therefore, 9 = Nine (four letters) => 4.

Therefore the missing number is 4.

Solution:

Take both the ropes. Now light the first rope from both the ends. Light the second rope from just one end. In thirty minutes, the first rope will burn fully. At this point of time, the second rope will be left exactly half and you have to light it from the second end now. It will take half of the actual time (thirty minutes) to burn now i.e. fifteen minute.

Thirty Minutes + Fifteen minutes = Forty Minutes

Solution:

The three numbers are:
-3, -1 and 1.

Solution:

The figure shows how we have done it. If you have any other idea in mind, feel free to comment below.

Solution:

No

Explanation:
7 is the only prime followed by a cube.
Assume that n3-1 is a prime for some n.
N3-1 will be equal to (n−1)(n2+n+1)
Now, n-1 divides n3-1.
If n-1>1, then we will have a contradiction to n3-1 being a prime.

Solution:

See the colored sticks in the picture. We have moved those sticks in the position given in the second image corresponding with the color.

Solution:

The wise man told them to switch the horses and then race. In this manner, both of them will be willing to race as fast as they can to prove that the other horse (his own horse) is slow.