work 021 – Own Inequalities

1) For a,b,c positive reals with sum 1 , show that –
\frac {a}{4b + 3bc + 4c} + \frac {b}{4c + 3ca + 4a} + \frac {c}{4a + 3ab + 4b} \ge \frac {1}{3}

2) For a,b,c positive reals with sum 3 , and the sum of any 2 greater than 1, show that –
\frac {a}{b + c - 1} + \frac {b}{c + a - 1} + \frac {c}{a + b - 1} \ge 3

3) For a,b,c positive reals with sum 3 , show that –
\frac {a}{2b + 3c - 1} + \frac {b}{2c + 3a - 1} + \frac {c}{2a + 3b - 1} \ge \frac {3}{4}

4) For x,y,z positive reals such that –

xy + y \ge 1

yz + z \ge 1

zx + x \ge 1

Prove that –
\frac {x}{1 + xz - x} + \frac {y}{1 + yx - y} + \frac {z}{1 + zx - z} + \frac {(x + y - 1)(z + zx - 1)(x + xz - 1)}{xyz} \ge 4

Proof

For the first three, use Cauchy-Scwarz in the Engel Form.

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