Problem 6

Let

n\ge3

be a positive integer. Let

C_1,C_2,\ldots,C_n

be unit circles in the plane, with centers

O_1,O_2,\ldots,O_n

respectively. If no line meets more than two of the circles, prove that

\sum_{1\le i<j\le n}\frac1{O_iO_j}\le\frac{(n-1)\pi}{4}

Step 1 of 3: Measure the tangent arcs

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In plain words

A tangent to one unit circle cuts the other circle only for a short pair of arcs.

\sin\frac{\theta_{ij}}2=\frac1{O_iO_j}

Detailed analysis

First, any two circles are disjoint. Indeed, if two met at a point, choose a line through that point and any point of a third circle; it would meet three circles. Fix $i\ne j$ . Let $\theta_{ij}$ be the angular length of each of the two arcs on $C_i$ whose tangent line meets $C_j$ . At an endpoint of such an arc, the distance from $O_j$ to the tangent is $1$ . The tangent point is displaced by half the arc length from the direction perpendicular to $O_iO_j$ , so the right-triangle relation is $1=O_iO_j\sin(\theta_{ij}/2)$ , namely $\sin(\theta_{ij}/2)=1/O_iO_j$ . Hence $1/O_iO_j=\sin(\theta_{ij}/2)\le\theta_{ij}/2$ .