- Find all positive integers $n$ such that there exist prime numbers $p, q$ satisfying $$1 !+2 !+3 !+\ldots+n !=p^{2}+q^{2}+5895$$
*Notice that $n !=1.2 .3 \ldots n$.* - Given an acute angle $A B C$ with the altitude $A H,$ the median $B M,$ the angle bisector $C K$. Show that if $HMK$ is an equilateral triangle then so is $A B C$.
- Suppose that $n$ is a positive integer so that $3^{n}+7^{n}$ is divisible by $11$. Find the remainder in the divison of $2^{n}+17^{n}+2018^{n^{2}}$ by $11$
- Given a circle $(O)$ and suppose that $B C$ is a fixed chord of $(O) .$ Let $A$ be a point moving on the major arc $B C ;$ and $M,$ $N$ respectively the midpoints of $A B$ and $A C$. Show that each of the altitudes $M M^{\prime}$ and $N N^{\prime}$ of $\triangle A M N$ contains some fixed point.
- Solve the equation $$13 x^{2}-5 x-13=(16 x-11) \sqrt{2 x^{2}-3}$$
- Solve the system of equations $$\begin{cases}\sqrt{4 x^{2}+5}+\sqrt{4 y^{2}+5}&=6|x y| \\ 2^{\dfrac{1}{x^{8}}+\dfrac{2}{y^{4}}}+2^{\dfrac{1}{y^{8}}+\dfrac{2}{x^{4}}}&=16 \end{cases}.$$
- Show that the following inequality holds for all positive numbers $a, b, c$ $$a^{2}+b^{2}+c^{2} \geq \frac{1}{2}(a b+b c+c a) + \sqrt{\frac{2(a+b+c)\left(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}\right)}{(a+b)(b+c)(c+a)}}$$
- Given triangle $A B C$ with the incenter $I$, the centroid $G$. There lines $A G$, $B G$ and $CG$ respectively intersect the circumcircle of $A B C$ at $A_{2}$, $B_{2}$, $C_{2}$. Show that $$G A_{2}+G B_{2}+G C_{2} \geq I A+I B+I C.$$
- Given continuous functions $f,g:[a, b] \rightarrow[a, b]$ such that $$f(g(x))=g(f(x)) \quad \forall x \in[a, b]$$ where $a, b$ are real numbers. Show that the equation $f(x)=g(x)$ has at least one solution.
- Find all pairs of integers $(m, n)$ so that both $m^{5} n+2019$ and $m n^{5}+2019$ are cubes of integers.
- Let $\quad S=\{1,2,3, \ldots, 2019\}$ and suppose that $S_{1}, S_{2}, \ldots, S_{410}$ are subsets of $S$ such that for every $i \in S$, there exist exactly $40$ of them containing i. Prove that there exist different indices $j, k, l \in\{1,2, \ldots, 410\}$ such that $\left|S_{j} \cap S_{k} \cap S_{l}\right| \leq 1$
- Given a cyclic quadrilateral $A B C D$ with $E$ is the intersection between $A C$ and $B D .$ The circumcircles of $E A D$ and $E B C$ intersect at another point $F$ (besides $E$). The perpendicular bisectors of $B D$, $A C$ respectively intersect $F B$, $F A$ at $K$, $L$. Show that $K L$ goes through the midpoint of $A B$.