OlympiadBench / 2082 / Find the largest possible integer k, such that the following statement is…

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 colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain

$$
\begin{aligned}
b_{1} \leq b_{2} \leq \ldots \leq b_{2009} & \text { the lengths of the blue sides } \\
r_{1} \leq r_{2} \leq \ldots \leq r_{2009} & \text { the lengths of the red sides, } \\
\text { and } \quad & w_{1} \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { the lengths of the white sides. }
\end{aligned}
$$

Then there exist k indices j such that we can form a non-degenerated triangle with side lengths b_(j),r_(j),w_(j).

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 colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain

$$
\begin{aligned}
b_{1} \leq b_{2} \leq \ldots \leq b_{2009} & \text { the lengths of the blue sides } \\
r_{1} \leq r_{2} \leq \ldots \leq r_{2009} & \text { the lengths of the red sides, } \\
\text { and } \quad & w_{1} \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { the lengths of the white sides. }
\end{aligned}
$$

Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_{j}, r_{j}, w_{j}$.