Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \]Prove that (i) $\left\{x_n\right\}$ is eventually periodic. (ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.

Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI$, $AC$ and $CI$ by $L$, $M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively. Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ$, $LN$ and $EF$ are concurrent.

Let \(a_1, a_2, \ldots, a_n\) be integers such that \(a_1 > a_2 > \cdots > a_n > 1\). Let \(M = \operatorname{lcm} \left( a_1, a_2, \ldots, a_n \right)\). For any finite nonempty set $X$ of positive integers, define \[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \]Such a set $X$ is called minimal if for every proper subset $Y$ of it, $f(Y) < f(X)$ always holds. Suppose $X$ is minimal and $f(X) \geqslant \frac{2}{a_n}$. Prove that \[ |X| \leqslant f(X) \cdot M. \]

The fractional distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined as \[ \sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2},\]where $\left\| x \right\|$ denotes the distance between $x$ and its nearest integer. Find the largest real $r$ such that there exists four points on the plane whose pairwise fractional distance are all at least $r$.

Let $p$ be a prime number and $f$ be a bijection from $\left\{0,1,\ldots,p-1\right\}$ to itself. Suppose that for integers $a,b \in \left\{0,1,\ldots,p-1\right\}$, $|f(a) - f(b)|\leqslant 2024$ if $p \mid a^2 - b$. Prove that there exists infinite many $p$ such that there exists such an $f$ and there also exists infinite many $p$ such that there doesn't exist such an $f$.

Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$. (i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \](ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).