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(105) 5

Part of 1985 Tournament Of Towns

Problems(1)

TOT 105 1985 Autumn S5 at least n- 1 acute angles <A_iOA_j

Source:

8/26/2019
(a) The point OO lies inside the convex polygon A1A2A3...AnA_1A_2A_3...A_n . Consider all the angles AiOAjA_iOA_j where i,ji, j are distinct natural numbers from 11 to nn . Prove that at least n−1n- 1 of these angles are not acute .
(b) Same problem for a convex polyhedron with nn vertices.
(V. Boltyanskiy, Moscow)
geometryconvexconvex polygonacuteconvex polyhedron3D geometry