MathLabs

Problem 2

Five points A1,A2,A3,A4,A5A_1,A_2,A_3,A_4,A_5 lie in the plane, with no three collinear. Determine the maximum possible value of the minimum angle ∠AiAjAk\angle A_iA_jA_k over all distinct i,j,ki,j,k.
Step 3 of 5: Place remaining points
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∠A1A2A3≤108∘\angle A_1A_2A_3\le108^\circ
Detailed analysis

The other two points lie inside the angle at A2A_2. Among the three rays from A2A_2 to A1,A3A_1,A_3 and the two remaining points, two consecutive rays have angle at most 108∘/3=36∘108^\circ/3=36^\circ.