$\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{\mathrm{n}}$ are points in the plane and $\mathrm{r}_{1}, \mathrm{r}_{2}, \ldots, \mathrm{r}_{\mathrm{n}}$ are real numbers such that the distance between $P_{i}$ and $P_{j}$ is $r_{i}+r_{j}$ (for $i$ not equal to $j$ ). Find the largest $n$ for which this is possible.
Answer: $\mathrm{n}=4$.

Draw a circle of radius $r_{i}$ at $P_{i}$. Then each pair of circles must touch. But that is possible if and only if $n \leq 4$.

![](https://cdn.mathpix.com/cropped/2024_05_09_1d234a78607cd781abe6g-34.jpg?height=477&width=557&top_left_y=2126&top_left_x=247)