2009 IMO Shortlist A1

CZE 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 $$\begin{aligned} & b_{1} \leq b_{2} \\ & \leq \ldots \leq b_{2009} \quad \text { the lengths of the blue sides, } \\ r_{1} & \leq r_{2} \leq \ldots \leq r_{2009} \quad \text { the lengths of the red sides, } \\ \text { and } \quad w_{1} & \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { the lengths of the white sides. } \end{aligned}$$ Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_{j}, r_{j}, w_{j}$ 。

CZE 找到最大可能的整数 $k$，使得以下陈述成立：给定 2009 个任意非退化三角形。每个三角形的三个边都有颜色，一条是蓝色，一条是红色，一条是白色。现在，对于每种颜色，让我们分别对边的长度进行排序。我们得到 $$\begin{aligned} & b_{1} \leq b_{2} \\ & \leq \ldots \leq b_{2009} \quad \text { 蓝色边的长度， } \\ r_{1} & \leq r_{2} \leq \ldots \leq r_{2009} \quad \text { 红色边的长度， } \\ \text {和 } \quad w_{1} & \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { 白边的长度。 } \end{aligned}$$ 那么存在$k$个索引$j$，这样我们就可以形成一个边长为$b_{j}, r_{j}, w_{j}$的非退化三角形。