Školjka

%V0 Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain $$b_1 \leqslant b_2 \leqslant \cdots \leqslant b_{2009} \qquad \text{the lengths of the blue sides,}$$ $$r_1 \leqslant r_2 \leqslant \cdots \leqslant r_{2009} \qquad \text{the lengths of the red sides,}$$ $$w_1 \leqslant w_2 \leqslant \cdots \leqslant w_{2009} \qquad \text{the lengths of the white sides.}$$ Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. Proposed by Michal Rolinek, Czech Republic