[b]p22.[/b] Consider the series $\{A_n\}^{\infty}_{n=0}$, where $A_0 = 1$ and for every $n > 0$, $$A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]},$$  where $[x]$ denotes the largest integer value smaller than or equal to $x$. Find the $(2023^{3^2}+20)$-th element of the series.


[b]p23.[/b] The side lengths of triangle $\vartriangle ABC$ are $5$, $7$ and $8$. Construct equilateral triangles $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$ such that $A_1$,$B_1$,$C_1$ lie outside of $\vartriangle ABC$. Let $A_2$,$B_2$, and $C_2$ be the centers of $\vartriangle A_1BC$, $\vartriangle B_1CA$, and $\vartriangle C_1AB$, respectively. What is the area of $\vartriangle A_2B_2C_2$?


[b]p24. [/b]There are $20$ people participating in a random tag game around an $20$-gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the $20$-gon (no matter where they were at the beginning). If there are currently $10$ taggers, let $E$ be the expected number of untagged people at the end of the next round. If $E$ can be written as $\frac{a}{b}$ for $a, b$ relatively prime positive integers, compute $a + b$.


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the $(2023^{3^2} + 20)$-th element of the series $\{A_n\}_{n=0}^{\infty}$, where $A_0 = 1$ and for $n > 0$,
\[ A_n = A_{\left\lfloor \frac{n}{2023} \right\rfloor} + A_{\left\lfloor \frac{n}{2023^2} \right\rfloor} + A_{\left\lfloor \frac{n}{2023^3} \right\rfloor}. \]

1. **Understanding the Recurrence Relation:**
   The recurrence relation given is:
   \[ A_n = A_{\left\lfloor \frac{n}{2023} \right\rfloor} + A_{\left\lfloor \frac{n}{2023^2} \right\rfloor} + A_{\left\lfloor \frac{n}{2023^3} \right\rfloor}. \]
   This means that the value of $A_n$ depends on the values of $A$ at indices that are significantly smaller than $n$.

2. **Base Case:**
   We are given that $A_0 = 1$.

3. **Pattern Recognition:**
   We need to recognize a pattern in the values of $A_n$. Let's consider the number of digits $s$ in the base-2023 representation of $n$. If $n$ has $s$ digits in base-2023, then:
   \[ A_n = f(s), \]
   where $f(s)$ is a function that depends on the number of digits $s$.

4. **Recurrence for $f(s)$:**
   From the given recurrence relation, we can derive a recurrence for $f(s)$:
   \[ f(s) = f(s-1) + f(s-2) + f(s-3) \quad \text{for} \quad s \geq 3. \]

5. **Initial Values:**
   We need the initial values of $f(s)$:
   \[ f(0) = 1 \quad (\text{since } A_0 = 1), \]
   \[ f(1) = 3 \quad (\text{since } A_1 = A_0 + A_0 + A_0 = 3), \]
   \[ f(2) = 5 \quad (\text{since } A_2 = A_0 + A_0 + A_1 = 5). \]

6. **Finding $f(10)$:**
   We need to find $f(10)$ using the recurrence relation:
   \[ f(3) = f(2) + f(1) + f(0) = 5 + 3 + 1 = 9, \]
   \[ f(4) = f(3) + f(2) + f(1) = 9 + 5 + 3 = 17, \]
   \[ f(5) = f(4) + f(3) + f(2) = 17 + 9 + 5 = 31, \]
   \[ f(6) = f(5) + f(4) + f(3) = 31 + 17 + 9 = 57, \]
   \[ f(7) = f(6) + f(5) + f(4) = 57 + 31 + 17 = 105, \]
   \[ f(8) = f(7) + f(6) + f(5) = 105 + 57 + 31 = 193, \]
   \[ f(9) = f(8) + f(7) + f(6) = 193 + 105 + 57 = 355, \]
   \[ f(10) = f(9) + f(8) + f(7) = 355 + 193 + 105 = 653. \]

Therefore, the $(2023^{3^2} + 20)$-th element of the series is $653$.

The final answer is $\boxed{653}$.