Prove that there are infinitely many positive integers $a$ such that \[a!+(a+2)!\mid (a+2\left\lfloor\sqrt{a}\right\rfloor)!.\] Proposed by Navid and the4seasons.

Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has $$ f(A)+f(C)=f(B)+f(D). $$ Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle. Proposed by Navid.

Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights. Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights. Proposed by usjl

On a connected graph $G$, one may perform the following operations: choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours choose a vertice $v$ with odd degree and delete it Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by idonthaveanaopsaccount

$I,\Omega$ are the incenter and the circumcircle of triangle $ABC$, respectively, and the tangents of $B,C$ to $\Omega$ intersect at $L$. Assume that $P\neq C$ is a point on $\Omega$ such that $CI,AP$, and the circle with center $L$ and radius $LC$ are concurrent. Let the foot from $I$ to $AB$ be $F$, the midpoint of $BC$ be $M$, $X$ is a point on $\Omega$ s.t. $AI,BC,PX$ are concurrent. Prove that the lines $AI,AX,MF$ form an isosceles triangle. Proposed by ckliao914

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) Proposed by the4seasons.