Problem 381

Since the average of the four numbers $4, a, b, 22$ is 13, then $\dfrac{4+a+b+22}{4}=13$ and so $4+a+b+22=52$ or $a+b=26$.

Since $a > 4$ and $a$ is an integer, then $a \geq 5$.

Since $a+b=26$ and $a<b$, then $a$ is less than half of 26, or $a<13$.

Since $a$ is an integer, then $a \leq 12$.

Therefore, we have $5 \leq a \leq 12$.

There are 8 choices for $a$ in this range: 5, 6, 7, 8, 9, 10, 11, 12. (Note that $12-5 + 1 = 8$.)

These give the pairs $(a,b)=(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14)$.

Thus, there are 8 possible pairs.