Question

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 = c^3$. Prove that $a^2 + b^2 - c^2 \gt 6(c - a)(c - b)$ .

score 133 · Answer 1 · 2017-09-17 22:50:37

133

Views

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 = c^3$. Prove that $a^2 + b^2 - c^2 \gt 6(c - a)(c - b)$ .

Published on 17 Sep 2017 - 22:50

score 226 · Answer 2 · 2017-09-19 04:17:20

226

Views

If $a,b,c$ are positive reals, then prove that $(a+b)(a+c)\ge 2 \sqrt {abc(a+b+c)}$

Published on 19 Sep 2017 - 4:17

score 1.2k · Answer 3 · 2026-03-30 04:44:25

1.2k

Views

Calculating area of quadrilateral when distance of vertices from an arbitrary point is known

Published on 30 Mar 2026 - 4:44

score 153 · Answer 4 · 2026-03-25 13:53:21

153

Views

Inequality on tangent and secant function

Published on 25 Mar 2026 - 13:53

score 650 · Answer 5 · 2026-03-25 20:20:49

650

Views

If $a$ and $b$ are positive real numbers such that $a+b=1$, then prove that $\big(a+\frac{1}{a}\big)^2 +\big(b+\frac{1}{b}\big)^2 \ge\frac{25}{2}$

Published on 25 Mar 2026 - 20:20

score 107 · Answer 6 · 2017-09-22 03:40:51

107

Views

$ \left( 1 - \frac{x}{n}\right)^n \leq \left( 1 - \frac{x}{n+1}\right)^{n+1} $, $\forall$ $x$ $\in$ $[0,n]$

Published on 22 Sep 2017 - 3:40

score 121 · Answer 7 · 2017-09-24 18:16:48

121

Views

If $x,y,z$ are positive numbers, then prove that $\frac {x}{x+y} + \frac{y}{y+z} +\frac {z}{z+x} \le 2$

Published on 24 Sep 2017 - 18:16

score 536 · Answer 8 · 2026-02-23 07:18:19

536

Views

Converse to the Duality Theorem of Geometric Programming

Published on 23 Feb 2026 - 7:18

score 70 · Answer 9 · 2017-09-25 05:42:22

70

Views

Convexity argument to maximize $\sqrt{\alpha \beta} + \sqrt{(1-\alpha)(1-\beta)}$ over $\alpha,\beta\in [0,1]$

Published on 25 Sep 2017 - 5:42

score 452 · Answer 10 · 2026-03-25 19:06:29

452

Views

$(xy+yz+zx)(1+6xyz) \geq 11xyz$

Published on 25 Mar 2026 - 19:06

score 230 · Answer 11 · 2017-09-27 17:07:40

230

Views

Seeking the maximal parameter value s.t. two-variable inequality still holds

Published on 27 Sep 2017 - 17:07

score 159 · Answer 12 · 2017-09-28 05:45:49

159

Views

Prove that $\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$

Published on 28 Sep 2017 - 5:45

score 198 · Answer 13 · 2017-09-30 12:56:23

198

Views

If we know range of a function , how can construct range of other function?

Published on 30 Sep 2017 - 12:56

score 93 · Answer 14 · 2017-10-01 01:43:30

93

Views

Prove that $\frac{1}{a^{3}(b+c)} +\frac{1}{b^{3}(a+c)} +\frac{1}{c^{3}(a+b)} \ge \frac{3}{2}$

Published on 01 Oct 2017 - 1:43

score 173 · Answer 15 · 2017-10-01 08:32:27

173

Views

A Hard Inequality

Published on 01 Oct 2017 - 8:32

List Question

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 = c^3$. Prove that $a^2 + b^2 - c^2 \gt 6(c - a)(c - b)$ .

If $a,b,c$ are positive reals, then prove that $(a+b)(a+c)\ge 2 \sqrt {abc(a+b+c)}$

Calculating area of quadrilateral when distance of vertices from an arbitrary point is known

Inequality on tangent and secant function

If $a$ and $b$ are positive real numbers such that $a+b=1$, then prove that $\big(a+\frac{1}{a}\big)^2 +\big(b+\frac{1}{b}\big)^2 \ge\frac{25}{2}$

$ \left( 1 - \frac{x}{n}\right)^n \leq \left( 1 - \frac{x}{n+1}\right)^{n+1} $, $\forall$ $x$ $\in$ $[0,n]$

If $x,y,z$ are positive numbers, then prove that $\frac {x}{x+y} + \frac{y}{y+z} +\frac {z}{z+x} \le 2$

Converse to the Duality Theorem of Geometric Programming

Convexity argument to maximize $\sqrt{\alpha \beta} + \sqrt{(1-\alpha)(1-\beta)}$ over $\alpha,\beta\in [0,1]$

$(xy+yz+zx)(1+6xyz) \geq 11xyz$

Seeking the maximal parameter value s.t. two-variable inequality still holds

Prove that $\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$

If we know range of a function , how can construct range of other function?

Prove that $\frac{1}{a^{3}(b+c)} +\frac{1}{b^{3}(a+c)} +\frac{1}{c^{3}(a+b)} \ge \frac{3}{2}$

A Hard Inequality

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