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Disk inequality

Source: ISL 2020 G4

July 20, 2021

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Problem Statement

In the plane, there are

n \geqslant 6

pairwise disjoint disks

D_{1}, D_{2}, \ldots, D_{n}

with radii

R_{1} \geqslant R_{2} \geqslant \ldots \geqslant 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} \geqslant R_{6}+R_{7}+\ldots+R_{n}.

(A disk is assumed to contain its boundary.)

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