# Own Inequalities

Problem 1

Show that $\forall$ positive reals $x,y,z$ such that,$x \ge y \ge z$,
$x \cdot (y^3) \ge 16(x - y)(y^2 - zx)$

Problem 2

Prove that the following inequalities hold for all  a,b,c – sides of a triangle
$\frac {c}{3a - b + c} + \frac {a}{3b - c + a} + \frac {b}{3c - a + b} \ge 1$

Problem 3

Prove that for positive reals $p,q,r$ positive reals,
$\sum_{cyclic}\frac {p^2}{q^2} > \sum_{cyclic}\frac {r}{p + q}$

Problem 4.

Prove that for positive reals a,b,c the following inequality holds good –
$a^2(\frac {a + 2c}{3b}) + b^2(\frac {b + 2a}{3c}) + c^2(\frac {c + 2b}{3a}) \ge a^2 + b^2 + c^2$

Problem 5.

Prove that for positive reals $x,y,z$ the following inequality holds –
$(x^2 - yz)^2 \ge \frac {27}{8}xy(xy - z^2)(xz - y^2)$

Problem 6.

Prove that for positive reals $a,b,c$ such that $abc = 1$,
the following inequality holds-
$\frac {a^3 + b^3 + c^3 + 3}{4} \ge \frac {b}{a + c} + \frac {c}{a + b} + \frac {a}{b + c}$

Problem 7

Prove that for positive reals $a,b,c$ with sum 2 ,
$\frac {a}{b(a + b)} + \frac {b}{c(b + c)} + \frac {c}{a(a + c)} \ge 2$

Problem 8.

Prove that for positive reals $a,b,c$ ,
$\frac {a^2 + ac}{2b + a + c} + \frac {b^2 + ba}{2c + a + b} + \frac {c^2 + cb}{2a + b + c} \ge \frac{1}{2}\cdot(a + b + c)$

Problem 9.

Let $a,b,c \in \mathbb{R}^ +$,
Prove that ,
$\frac {a^2 + ab + b^2}{(a + b)^2} + \frac {b^2 + bc + c^2}{(b + c)^2} + \frac {c^2 + ca + a^2}{(c + a)^2} \ge \frac {9}{4}$

Problem 10.

Let function f be defined such that –
$f(c) = (b - c)(a + c) + c^2$ where a,b,c are positive reals and a,b are fixed with $a\ge b$. Pove that the following inequality holds true –
$f(a - b) + f(c) \le a^2 + b^2$

Problem 11.

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

Problem 12.

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

Problem 13.

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

Problem 14.

For $x,y,z$ positive reals such that – $xy + y \ge 1 , yz + z \ge 1 , zx + x \ge 1$ ,

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

Problem 15.

For $x,y,z \in \mathbb{R}^+$ with $xy+yz+zx=1$ , Prove that –

$15x^2 + 7y^2 + 3z^2 \ge 6$

Problem 16.

For $x,y,z \in \mathbb{R}^+$ such that $x^4+y^4+z^4=3$. Prove that-

$x^\frac{25}{6} + y^\frac{25}{6} + z^\frac{25}{6} \ge 3$