[b]p7.[/b] An ant starts at the point $(0, 0)$. It travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. If the ant reaches $(20, 23)$, what is the probability it did not pass through $(20, 20)$?


[b]p8.[/b] Let $a_0 = 2023$ and $a_n$ be the sum of all divisors of $a_{n-1}$ for all $n \ge 1$. Compute the sum of the prime numbers that divide $a_3$.


[b]p9.[/b] Five circles of radius one are stored in a box of base length five as in the following diagram. How far above the base of the box are the upper circles touching the sides of the box?
[img]https://cdn.artofproblemsolving.com/attachments/7/c/c20b5fa21fbd8ce791358fd888ed78fcdb7646.png[/img]


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to compute the sum of the prime numbers that divide \(a_3\). We start with \(a_0 = 2023\) and use the given recurrence relation \(a_n\) which is the sum of all divisors of \(a_{n-1}\).

1. **Compute \(a_1\):**
   \[
   a_0 = 2023 = 7 \cdot 17^2
   \]
   The sum of the divisors of \(2023\) is:
   \[
   a_1 = (1 + 7)(1 + 17 + 17^2) = 8 \cdot (1 + 17 + 289) = 8 \cdot 307 = 2456
   \]

2. **Compute \(a_2\):**
   \[
   2456 = 2^3 \cdot 307
   \]
   The sum of the divisors of \(2456\) is:
   \[
   a_2 = (1 + 2 + 4 + 8)(1 + 307) = 15 \cdot 308 = 4620
   \]

3. **Compute \(a_3\):**
   \[
   4620 = 2^2 \cdot 3 \cdot 5 \cdot 7 \cdot 11
   \]
   The sum of the divisors of \(4620\) is:
   \[
   a_3 = (1 + 2 + 4)(1 + 3)(1 + 5)(1 + 7)(1 + 11)
   \]
   Calculate each term:
   \[
   1 + 2 + 4 = 7
   \]
   \[
   1 + 3 = 4
   \]
   \[
   1 + 5 = 6
   \]
   \[
   1 + 7 = 8
   \]
   \[
   1 + 11 = 12
   \]
   Therefore:
   \[
   a_3 = 7 \cdot 4 \cdot 6 \cdot 8 \cdot 12
   \]
   Simplify the product:
   \[
   a_3 = 7 \cdot (2^2) \cdot (2 \cdot 3) \cdot (2^3) \cdot (2^2 \cdot 3) = 2^8 \cdot 3^2 \cdot 7
   \]

4. **Sum of the prime numbers that divide \(a_3\):**
   The prime factors of \(a_3\) are \(2\), \(3\), and \(7\). Their sum is:
   \[
   2 + 3 + 7 = 12
   \]

The final answer is \(\boxed{12}\)