Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.

Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$ with $AB\neq AC$, respectively. Let $M$ be the midpoint of $BC$ and $K$ be the intersection of $AM$ and the circumcircle of $\triangle BHC$, such that $M$ lies between $A$ and $K$. Let $N$ be the intersection of $HK$ and $BC$. Show that if $\angle BAM=\angle CAN$, then $AN\perp OH$.

Let $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$ be a set consisting of points on the coordinate plane. Each element of $S$ is colored one of four given colors. A subset $T$ of $S$ is called colorful if $T$ consists of exactly $4$ points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets $S$ can have, among all legitimate coloring patters.

Let $n$ be a given integer such that $n\ge 2$. Find the smallest real number $\lambda$ with the following property: for any real numbers $x_1,x_2,\ldots ,x_n\in [0,1]$ , there exists integers $\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\}$ such that the inequality $$\left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambda$$holds for all pairs of integers $(i,j)$ where $1\le i\le j\le n$.

In acute-angled triangle $ABC,$ $AB>AC.$ Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. The line passing through $H$ and parallel to $AB$ intersects line $AC$ at $M,$ and the line passing through $H$ and parallel to $AC$ intersects line $AB$ at $N.$ $L$ is the reflection of the point $H$ in $MN.$ Line $OL$ and $AH$ intersect at $K.$ Prove that $K,M,L,N$ are concyclic.

Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive numbers such that $a_1\leq a_2 \leq \cdots \leq a_n .$ Prove that $$\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.$$

Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

We call a set $S$ a good set if $S=\{x,2x,3x\}(x\neq 0).$ For a given integer $n(n\geq 3),$ determine the largest possible number of the good subsets of a set containing $n$ positive integers.