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 2 of 5: Choose an extreme angle
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convex hull\text{convex hull}
Detailed analysis

For any five points, their convex hull is a triangle, quadrilateral, or pentagon. It has an interior angle at most 108∘108^\circ. Relabel so this angle is ∠A1A2A3\angle A_1A_2A_3.