ELMO Shortlist 2012 C8

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair (x,y) denote the complex number x+y\omega for \omega=e^{2\pi i/3}. We define an \omega-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form x=a or y=b, where a and b are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an \omega-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a tasteful tiling is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).

a) Prove that if an \omega-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.

b) Prove that such a tasteful tiling is unique.


ELMO Shortlist 2012 C9

For a set A of integers, define f(A)=\{x^2+xy+y^2: x,y\in A\}. Is there a constant c such that for all positive integers n, there exists a set A of size n such that |f(A)|\le cn?