MathLabs

Problem 6

Let n≥3n\ge3 be a positive integer. Let C1,C2,…,CnC_1,C_2,\ldots,C_n be unit circles in the plane, with centers O1,O2,…,OnO_1,O_2,\ldots,O_n respectively. If no line meets more than two of the circles, prove that ∑1≤i<j≤n1OiOj≤(n−1)π4\sum_{1\le i<j\le n}\frac1{O_iO_j}\le\frac{(n-1)\pi}{4}.
Step 3 of 3: Finish by summing
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In plain words

The total available angular length is at most 2(n−1)π2(n-1)\pi.

8∑i<j1OiOj≤2(n−1)π8\sum_{i<j}\frac1{O_iO_j}\le2(n-1)\pi
Detailed analysis

For every pair, the first step gives 8/OiOj=8sin⁡(θij/2)≤4θij8/O_iO_j=8\sin(\theta_{ij}/2)\le4\theta_{ij}. Summing over all pairs and using the definition of αi\alpha_i, we obtain 8∑i<j1OiOj≤4∑i<jθij=∑iαi≤2(n−1)π8\sum_{i<j}\frac1{O_iO_j}\le4\sum_{i<j}\theta_{ij}=\sum_i\alpha_i\le2(n-1)\pi. Dividing by 88 yields ∑i<j1OiOj≤(n−1)π4\sum_{i<j}\frac1{O_iO_j}\le\frac{(n-1)\pi}{4}, as required.