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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
\[ \left. \begin{array}{rcl}
 & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }
 & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }
 \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }
 \end{array}\right.\]
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*

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