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pentagon inequality

Source: Izho 2015 problem 6

January 14, 2015
inequalitiesgeometrytrigonometryinequalities unsolved

Problem Statement

The area of a convex pentagon ABCDEABCDE is SS, and the circumradii of the triangles ABCABC, BCDBCD, CDECDE, DEADEA, EABEAB are R1R_1, R2R_2, R3R_3, R4R_4, R5R_5. Prove the inequality R14+R24+R34+R44+R54≥45sin⁡2108∘S2. R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2.
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