For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$. Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$

Find all integers $k$, such that there exists an integer sequence ${\{a_n\}}$ satisfies two conditions below (1) For all positive integers $n$,$a_{n+1}={a_n}^3+ka_n+1$ (2) $|a_n| \leq M$ holds for some real $M$

$AB,AC$ are tangent to $\Omega$ at $B$ and $C$, respectively. $D,E,F$ lie on segments $BC,CA,AB$ such that $AF<AE$ and $\angle FDB= \angle EDC$. The circumcircle of $\triangle FEC$ intersects $\Omega$ at $G$ and $C$. Show that $ \angle AEF= \angle BGD$

Given positive integer $n \geq 2$. Now Cindy fills each cell of the $n*n$ grid with a positive integer not greater than $n$ such that the numbers in each row are in a non-decreasing order (from left to right) and numbers in each column is also in a non-decreasing order (from top to bottom). We call two adjacant cells form a $domino$ , if they are filled with the same number. Now Cindy wants the number of $domino$s as small as possible. Find the smallest number of $dominos$ Cindy can reach. (Here, two cells are called adjacant if they share one common side)

Given hexagon $ \mathcal{P}$ inscribed in a unit square, such that each vertex is on the side of the square. It’s known that all interior angles of the hexagon are equal. Find the maximum possible value of the smallest side length of $\mathcal{P}$.

Alice and Bob now play a magic show. There are $101 $ different hats lie on the table and they form a circle. Firstly, Bob choose a positive integer $n$(Alice doesn’t know it). Then Bob puts a rabbit under one of the hats and Alice doesn’t know which hat contains the rabbit. Each time, she can choose a hat and see whether the rabbit is under the hat. If not, then Bob will move the rabbit from the current hat to the $n$th hat in a clockwise direction. They will repeat these steps until Alice find the rabbit. Prove that Alice can find the rabbit in $201$ steps.

Let $a,b,c,d$ be four positive integers such that $a>b>c>d$. Given that $ab+bc+ca+d^2|(a+b)(b+c)(c+a)$. Find the minimal value of $ \Omega (ab+bc+ca+d^2)$. Here $ \Omega(n)$ denotes the number of prime factors $n$ has. e.g. $\Omega(12)=3$

Given a positive integer $n \geq 2$. Let $a_{ij}$ $(1 \leq i,j \leq n)$ be $n^2$ non-negative reals and their sum is $1$. For $1\leq i \leq n$, define $R_i=max_{1\leq k \leq n}(a_{ik})$. For $1\leq j \leq n$, define $C_j=min_{1\leq k \leq n}(a_{kj})$ Find the maximum value of $C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)$