The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence: $$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$Given that $a_{2021} = 2021$, find $a_1$.

Let $P$, $A$, $B$ and $C$ be points on a line $r$, in that order, so that $AB = BC$. Let $H$ be a point that does not belong to this line and let $S$ be the other intersection of the circles $(HPB)$ and $(HAC)$. Let $I$ be the second intersection of the circle with diameter $HB$ and $(HAC)$. Show that the points $P$, $H$, $I$ lie on the same line if and only if $HS$ is perpendicular to $r$.

Let $n$ be a positive integer. In the $\mathit{philand}$ language, words are all finite sequences formed by the letters "$P$", "$H$" and "$I$". Philipe, who speaks only the $\mathit{philand}$ language, writes the word $PHIPHI\ldots PHI$ on a piece of paper, where $PHI$ is repeated $n$ times. He can do the following operations: • Erase two identical letters and write in their place two different letters from the original and from each other; (Ex: $PP\rightarrow HI$) • Erase two distinct letters and rewrite them changing the order in which they appear; (Ex: $PI\rightarrow IP$) • Erase two distinct letters and write the letter distinct from the two he erased. (Ex: $PH\rightarrow I$) Find the largest integer $C$ such that any Philandese word of up to $C$ letters can be written by Philip through the above operations. Note: Operations are taken on adjacent letters.

Let $H$ be the orthocenter of the triangle $ABC$ and let $D$, $E$, $F$ be the feet of heights by $A$, $B$, $C$. Let $\omega_D$, $\omega_E$, $\omega_F$ be the incircles of $FEH$, $DHF$, $HED$ and let $I_D$, $I_E$, $I_F$ be their centers. Show that $I_DD$, $I_EE$ and $I_FF$ compete.

Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$: $$f(mf(n)) + f(n) | mn + f(f(n)).$$