2006 IMO Shortlist C6

A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^{\circ}-120^{\circ}$ unit rhombus. Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1 \leq k \leq n$. (Colombia)

有孔三角形是边长 $n$ 的向上等边三角形，并切有 $n$ 向上的单位三角形孔。钻石是一个 $60^{\circ}-120^{\circ}$ 单位菱形。证明有孔三角形 $T$ 可以用菱形平铺，当且仅当以下条件成立：$T$ 中边长 $k$ 的每个向上等边三角形最多包含 $k$ 个孔，即 $1 \leq k \leq n$。 (哥伦比亚)