Let P, Q, and R be statement variables. Without using a truth table, and citing all properties used, prove that [PV ¬(¬Q V R)] = [(R ⇒ P) ^ (P V Q)]. Prove that for all a € Z, a³ - 6a² + 5a + 1 is odd. Prove that for all x € R, 1 + 99 sin² x ≥ 10 sin 2x.

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