# Math Help - Proof

1. ## Proof

Prove the following:

Assume that the average of four integers (all integers are different) is 20. Prove that at least 1 of those 4 integers has to be greater than 21.

2. Originally Posted by DiscreteW
Prove the following:

Assume that the average of four integers (all integers are different) is 20. Prove that at least 1 of those 4 integers has to be greater than 21.
Assume that they are all <= 21.

The biggest total possible would then be 21+20+19+18 so the biggest mean possible would be 19.5 but the mean is 20 so they are not all <=21 so at least one is greater than 21.

3. Proof:

We know that for average of 4 integers to be 20, the sum of those integers must be 80. With this in mind, we attempt to find the maximum sum of 4 different integers less than or equal to 21.

$
21 + 20 + 19 + 18 = 78$

and,

$78 < 80$

Therefore, there is no combination of 4 different integers less than or equal to 21 that have an average of 20.