Problem (Algebra):
For each positive real number $x$, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \leq p \leq x+10$. What is the value of $f(f(20))$ ?

Solution:
Let $a=f(20)$. Then $f(f(20))=f(a)$.

To calculate $f(f(20))$, we determine the value of $a$ and then the value of $f(a)$.

By definition, $a=f(20)$ is the number of prime numbers $p$ that satisfy $20 \leq p \leq 30$.

The prime numbers between 20 and 30, inclusive, are 23 and 29 , so $a=f(20)=2$.

Thus, $f(f(20))=f(a)=f(2)$.

By definition, $f(2)$ is the number of prime numbers $p$ that satisfy $2 \leq p \leq 12$.

The prime numbers between 2 and 12, inclusive, are $2,3,5,7,11$, of which there are 5 .

Therefore, $f(f(20))=5$.

Final answer: 5