Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ 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 $k$.
2018 Singapore MO Open
2nd Round
Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$. Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.
Let $n$ be a positive integer. Show that there exists an integer $m$ such that \[ 2018m^2+20182017m+2017 \]is divisible by $2^n$.
each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 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 $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$ Proposed by Ma Zhao Yu