Consider a regular cube with side length $2$. Let $A$ and $B$ be two vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1,A_2,\ldots,A_k$ so that $A_1=A$, $A_k=B$ and for any $i=1,\ldots,k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.
Let $O$ be a point inside a triangle $ABC$ such that $\angle BOC=90^\circ$ and $\angle BAO=\angle BCO$. Prove that $\angle OMN=90^\circ$, where $M,N$ are the midpoints of $AC$ and $BC$ respectively.
Each of the squares in a $2\times 2018$ grid of squares is to be coloured black or white such that in any $2\times 2$ block, at least one of the 4 squares is white. Let $P$ be the number of ways of colouring the grid. Find the largest $k$ so that $3^k$ divides $P$.
Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c$, $$P(a,b,c)=0\quad\Longleftrightarrow\quad a=b=c.$$ Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $n,m$, $$m^r\mid P(n,n+m,n+2 m).$$