Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions. $a_1=2,\,b_1=4,\,c_1=5$ $\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}$ Prove that for all positive integers $n$, $ $ $ $ $max(a_n,b_n,c_n)>\sqrt{2n+13}$.

In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.

For positive integers $m, n$ ($m>n$), $a_{n+1}, a_{n+2}, ..., a_m$ are non-negative integers that satisfy the following inequality. $$ 2> \frac{a_{n+1}}{n+1} \ge \frac{a_{n+2}}{n+2} \ge \cdots \ge \frac{a_m}{m}$$Find the number of pair $(a_{n+1}, a_{n+2}, \cdots, a_m)$.

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$.

$n(\geq 4)$ islands are connected by bridges to satisfy the following conditions: Each bridge connects only two islands and does not go through other islands. There is at most one bridge connecting any two different islands. There does not exist a list $A_1, A_2, \ldots, A_{2k}(k \geq 2)$ of distinct islands that satisfy the following: For every $i=1, 2, \ldots, 2k$, the two islands $A_i$ and $A_{i+1}$ are connected by a bridge. (Let $A_{2k+1}=A_1$) Prove that the number of the bridges is at most $\frac{3(n-1)}{2}$.

Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions: $a_i \leq a_j$ for every positive integers $i <j$. For any positive integer $k \geq 3$, the following inequality holds: $$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$ Prove that $\{a_n\}$ is constant.

$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation. $$p^2 x^4-6px^2+1=y^2$$