SMO 2019 Senior Round 2

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

Let $n,m$ be positive integers such that $n\ge2m+1$. Given $n$ distinct points and $nm$ distinct line segments, each joining a pair of points, prove that there exist distinct points $A_0,A_1,\ldots,A_m$, so that for each $i=0,\ldots,m-1$, there is a line segment joining $A_i$ to $A_{i+1}$.

Let $a_1,a_2,\ldots,a_{2000}$ be distinct positive integers such that $1\le a_1<a_2<\cdots<a_{2000}<4000$ such that the LCM (least common multiple) of any two of them is $\ge4000$. Prove that $a_1\ge1334$.

Let $m,n,k$ be positive integers so that $m\ge n$ and $1+2+\cdots+n=mk$. Show that the numbers $1,2,\ldots,n$ can be partitioned into $k$ groups so that the sum of the numbers in each of the groups is $m$.

Determine all integer $n\ge2$ such that it is possible to construct an $n\times n$ array where each entry is either $0$, $1$ or $-1$ so that the sums of elements in every row and every column are distinct.