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Question Number 144901 by loveineq last updated on 30/Jun/21

Let a, b > 0 and a + b + 1 = 3 a b . Prove that

\frac{a}{a^{2} + 1} + \frac{b}{b^{2} + 1} ⩽ 1 ⩽ \frac{a^{3}}{a^{2} + 1} + \frac{b^{3}}{b^{2} + 1}

Let a, b > 0, n \in Z^{+} and a + b + 1 = 3 a b . Prove or disprove

\frac{a^{n - 1}}{a^{n} + 1} + \frac{b^{n - 1}}{b^{n} + 1} ⩽ 1 ⩽ \frac{a^{n + 1}}{a^{n} + 1} + \frac{b^{n + 1}}{b^{n} + 1}