In a parallelogram $ABCD$, the bisector of $\angle A$ intersects $BC$ at $M$ and the extension of $DC$ at $N$. Let $O$ be the circumcircle of the triangle $MCN$. Prove that $\angle OBC = \angle ODC$

Graph $G$ has $n$ vertices and $mn$ edges, where $n>2m$, show that there exists a path with $m+1$ vertices. (A path is an open walk without repeating vertices )

Let $a_1,a_2,\cdots,a_{2000}$ be distinct positive integers such that $1 \leq a_1 < a_2 < \cdots < a_{2000} < 4000$ such that the LCM (least common multiple) of any two of them is $\geq 4000$. Show that $a_1 \geq 1334$

Positive integers $m,n,k$ satisfy $1+2+3++...+n=mk$ and $m \ge n$. Show that we can partite $\{1,2,3,...,n \}$ into $k$ subsets (Every element belongs to exact one of these $k$ subsets), such that the sum of elements in each subset is equal to $m$.

Determine all integer $n \ge 2$ such that it is possible to construct an $n * n$ array where each entry is either $-1, 0, 1$ so that the sums of elements in every row and every column are distinct