Average of 30 integers

Total sum of the 30 integers $= 30 \times 5 = 150$.

Let the 20 integers that do not exceed 5 have sum $S_{20}$, and the remaining 10 integers (each exceeding 5, so each $\ge 6$) have sum $S_{10}$.

To maximize the average of the 20 integers, we must maximize $S_{20}$, which means minimizing $S_{10}$.

Since each of the 10 integers is greater than 5, the minimum integer value for each is 6. $\text{Min } S_{10} = 10 \times 6 = 60$ $\text{Max } S_{20} = 150 - 60 = 90$

The highest possible average of these 20 integers is $\frac{90}{20} = 4.5$.