Are there different integers $a,b,c,d,e,f$ such that they are the $6$ roots of $$(x+a)(x^2+bx+c)(x^3+dx^2+ex+f)=0?$$

In a certain country there are $2023$ islands and $2022$ bridges, such that every bridge connects two different islands and any two islands have at most one bridge in common, and from any island, using bridges one can get to any other island. If in any three islands there is an island with bridges connected to each of the other two islands, call these three islands an "island group". We know that any two "island group"s have at least $1$ common island. What is the minimum number of islands with only $1$ bridge connected to it?

In $\triangle ABC$, points $P,Q$ satisfy $\angle PBC = \angle QBA$ and $\angle PCB = \angle QCA$, $D$ is a point on $BC$ such that $\angle PDB=\angle QDC$. Let $X,Y$ be the reflections of $A$ with respect to lines $BP$ and $CP$, respectively. Prove that $DX=DY$.

Let ${p}$ be a prime. $a,b,c\in\mathbb Z,\gcd(a,p)=\gcd(b,p)=\gcd(c,p)=1.$ Prove that: $\exists x_1,x_2,x_3,x_4\in\mathbb Z,| x_1|,|x_2|,|x_3|,|x_4|<\sqrt p,$ satisfying $$ax_1x_2+bx_3x_4\equiv c\pmod p.$$Proposed by Wang Guangting

Let $a_1,a_2,\cdots,a_{100}\geq 0$ such that $\max\{a_{i-1}+a_i,a_i+a_{i+1}\}\geq i $ for any $2\leq i\leq 99.$ Find the minimum of $a_1+a_2+\cdots+a_{100}.$

As shown in the figure, let point $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. The circumcenter of triangle $ABE$ is denoted as $K$. Point $X$ is the reflection of point $B$ with respect to the line $CD$, and point $Y$ is the point on the plane such that quadrilateral $DKEY$ is a parallelogram. Prove that the points $D,E,X,Y$ are concyclic.

For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

In a grid of $100\times 100$ squares, there is a mouse on the top-left square, and there is a piece of cheese in the bottom-right square. The mouse wants to move to the bottom-right square to eat the cheese. For each step, the mouse can move from one square to an adjacent square (two squares are considered adjacent if they share a common edge). Now, any divider can be placed on the common edge of two adjacent squares such that the mouse cannot directly move between these two adjacent squares. A placement of dividers is called "kind" if the mouse can still reach the cheese after the dividers are placed. Find the smallest positive integer $n$ such that, regardless of any "kind" placement of $2023$ dividers, the mouse can reach the cheese in at most $n$ steps.