A sequence of positive reals $\{ a_n \}$ is defined below. $$a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}$$Show that for all nonnegative integer $n$, $a_n$ is a positive integer.

Sets $A_0, A_1, \dots, A_{2023}$ satisfy the following conditions: $A_0 = \{ 3 \}$ $A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}$ for each $n = 1, 2, \dots, 2023$. Find $|A_{2023}|$.

For a given positive integer $n(\ge 2)$, find maximum positive integer $A$ such that there exists $P \in \mathbb{Z}[x]$ with degree $n$ that satisfies the following two conditions. For any $1 \le k \le A$, it satisfies that $A \mid P(k)$, and $P(0)= 0$ and the coefficient of the first term of $P$ is $1$, which means that $P(x)$ is in the following form where $c_2, c_3, \cdots, c_n$ are all integers and $c_n \neq 0$. $$P(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x$$

Pentagon $ABCDE$ is inscribed in circle $\Omega$. Line $AD$ meets $CE$ at $F$, and $P (\neq E, F)$ is a point on segment $EF$. The circumcircle of triangle $AFP$ meets $\Omega$ at $Q(\neq A)$ and $AC$ at $R(\neq A)$. Line $AD$ meets $BQ$ at $S$, and the circumcircle of triangle $DES$ meets line $BQ, BD$ at $T(\neq S), U(\neq D)$, respectively. Prove that if $F, P, T, S$ are concyclic, then $P, T, R, U$ are concyclic.

Find all positive integers $n$ such that $$\phi(n) + \sigma(n) = 2n + 8.$$

Let $\Omega$ and $O$ be the circumcircle and the circumcenter of an acute triangle $ABC$ $(\overline{AB} < \overline{AC})$. Define $D,E(\neq A)$ be the points such that ray $AO$ intersects $BC$ and $\Omega$. Let the line passing through $D$ and perpendicular to $AB$ intersects $AC$ at $P$ and define $Q$ similarly. Tangents to $\Omega$ on $A,E$ intersects $BC$ at $X,Y$. Prove that $X,Y,P,Q$ lie on a circle.

Positive real sequences $\{ a_n \}$ and $\{ b_n \}$ satisfy the following conditions for all positive integers $n$. $a_{n+1}b_{n+1}= a_n^2 + b_n^2$ $a_{n+1}+b_{n+1}=a_nb_n$ $a_n \geq b_n$ Prove that there exists positive integer $n$ such that $\frac{a_n}{b_n}>2023^{2023}.$

For a positive integer $n$, if $n$ is a product of two different primes and $n \equiv 2 \pmod 3$, then $n$ is called "special number." For example, $14, 26, 35, 38$ is only special numbers among positive integers $1$ to $50$. Prove that for any finite set $S$ with special numbers, there exist two sets $A, B$ such that $A \cap B = \emptyset, A \cup B = S$ $||A| - |B|| \leq 1$ For all primes $p$, the difference between number of elements in $A$ which is multiple of $p$ and number of elements in $B$ which is multiple of $p$ is less than or equal to $1$.