Squareland is a6-by 6 grid. Each cell is either a kingdom or a disputed territory; there are 27 kingdoms and 9 disputed territories. A kingdom claims a disputed cell exactly when it shares a side or a vertex with it. Can the nine disputed cells all have different numbers of claimants?
Let $ABCD$ be a rhombus and $APQC$ a parallelogram with $B$ inside it and $AP$ equal to a side of the rhombus. Prove that $B$ is the orthocentre of triangle $DPQ$.
Let $n$ be an integer. If $x^2+y^2+z^2-xy-yz-zx=n$ has an integer solution, prove that $u^2+v^2-uv=n$ also has an integer solution. (The source reuses the letters $x,y$ for the second equation.)
Two identical counters occupy a1 and c3 on an 8-by 8 chessboard. Alice and Bob alternate, Alice first. A move chooses either counter and moves it any positive number of squares rightwards or upwards. At most one counter can occupy a square, and counters cannot jump over each other. The player placing a counter on h8 wins. Who can force a win?
