Nemotron-IMO-Bench 2026 G21

In the plane, there are $n \geq 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geq R_{2} \geq \ldots \geq R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that $$O P_{1}+O P_{2}+\ldots+O P_{n} \geq R_{6}+R_{7}+\ldots+R_{n}.$$(A disk is assumed to contain its boundary.)