The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

Let the sum of the 30 integers be $S$. Since their average is 5:
$S=30\times 5=150$

Let the 20 integers that do not exceed 5 be $x_1,x_2,\dots ,x_{20}$. For each of these integers, $x_i\le 5$. Let their sum be $S_{20}$ and their average be $A_{20}=\frac{S_{20}}{20}$.

The remaining 10 integers must be strictly greater than 5, so each of them is at least 6. The minimum sum of these 10 integers is:
$S_{10}\ge 10\times 6=60$

Since the total sum is 150:
$S_{20}+S_{10}=150$
$S_{20}=150-S_{10}$

To maximize the sum (and thus the average) of the 20 integers, we must minimize $S_{10}$:
$S_{20,\text{max}}=150-60=90$

The highest possible average of these 20 integers is:
$A_{20,\text{max}}=\frac{90}{20}=4.5$