Question

$ 27 + ( 2 + \frac{a^{2}}{bc} ) (2 + \frac{b^{2}}{ac}) (2 + \frac{c^{2}}{ab}) \ge 6 (a+b+c)(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $

score 69 · Answer 1 · 2026-03-27 01:00:12

69

Views

$ 27 + ( 2 + \frac{a^{2}}{bc} ) (2 + \frac{b^{2}}{ac}) (2 + \frac{c^{2}}{ab}) \ge 6 (a+b+c)(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $

Published on 27 Mar 2026 - 1:00

score 147 · Answer 2 · 2019-11-27 04:45:58

147

Views

Cauchy-Schwarz inequality - proof

Published on 27 Nov 2019 - 4:45

score 92 · Answer 3 · 2019-11-28 00:38:36

92

Views

Determine the minimum of:$\frac{\sqrt{(a^2+1)(b^2+1)(c^2+1)(d^2+1)}}{a+b+c+d},\text{ if }a,b,c,d>0$

Published on 28 Nov 2019 - 0:38

score 60 · Answer 4 · 2019-12-01 07:10:48

60

Views

Alternative elementary method to find maximum

Published on 01 Dec 2019 - 7:10

score 20 · Answer 5 · 2019-12-02 02:07:20

20

Views

inequalities involving norm and probability

Published on 02 Dec 2019 - 2:07

score 300 · Answer 6 · 2026-03-23 03:52:03

300

Views

If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$

Published on 23 Mar 2026 - 3:52

score 110 · Answer 7 · 2026-03-25 12:50:05

110

Views

$a, b, c$ are three reals such that $ab + bc + ca = 3abc$. Prove that $\sum_{cyc}\sqrt{2(a + b)} \ge \sum_{cyc}\sqrt{\frac{a^2 + b^2}{a + b}} + 3$.

Published on 25 Mar 2026 - 12:50

score 191 · Answer 8 · 2019-12-11 12:06:09

191

Views

For positive $a$, $b$, $c$, show $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a} \ge \frac{a+b+c}{a+b+c+\sqrt[3]{abc}}$

Published on 11 Dec 2019 - 12:06

score 85 · Answer 9 · 2026-03-25 12:53:36

85

Views

Prove inequality by inequality of means

Published on 25 Mar 2026 - 12:53

score 167 · Answer 10 · 2026-03-27 22:31:07

167

Views

Maximum value of $\frac{x}{y+1}+\frac{y}{x+1}$ while $0\leq x,y \leq 1$

Published on 27 Mar 2026 - 22:31

score 107 · Answer 11 · 2026-03-27 02:36:08

107

Views

$\frac1{(a+1)^2}+ \frac1{(b+1)^2}\geq\frac 1{1+ab}$ for positive reals $a$ and $b$

Published on 27 Mar 2026 - 2:36

score 589 · Answer 12 · 2019-12-17 06:05:00

589

Views

Deriving Cauchy-Schwarz inequality for random variables

Published on 17 Dec 2019 - 6:05

score 292 · Answer 13 · 2019-12-18 18:25:45

292

Views

Proving $\frac{a}{a^2+2b^2}+\frac{b}{b^2+2c^2}+\frac{c}{c^2+2a^2}\geq 1$ when $a^2+b^2+c^2=a^3+b^3+c^3$

Published on 18 Dec 2019 - 18:25

score 89 · Answer 14 · 2019-12-19 17:00:26

89

Views

Show that $\sqrt{\langle T(x), x \rangle}$ is a norm when $T$ is positive definite

Published on 19 Dec 2019 - 17:00

score 159 · Answer 15 · 2019-12-20 09:34:33

159

Views

Find the maximum of a sum

Published on 20 Dec 2019 - 9:34

List Question

$ 27 + ( 2 + \frac{a^{2}}{bc} ) (2 + \frac{b^{2}}{ac}) (2 + \frac{c^{2}}{ab}) \ge 6 (a+b+c)(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $

Cauchy-Schwarz inequality - proof

Determine the minimum of:$\frac{\sqrt{(a^2+1)(b^2+1)(c^2+1)(d^2+1)}}{a+b+c+d},\text{ if }a,b,c,d>0$

Alternative elementary method to find maximum

inequalities involving norm and probability

If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$

$a, b, c$ are three reals such that $ab + bc + ca = 3abc$. Prove that $\sum_{cyc}\sqrt{2(a + b)} \ge \sum_{cyc}\sqrt{\frac{a^2 + b^2}{a + b}} + 3$.

For positive $a$, $b$, $c$, show $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a} \ge \frac{a+b+c}{a+b+c+\sqrt[3]{abc}}$

Prove inequality by inequality of means

Maximum value of $\frac{x}{y+1}+\frac{y}{x+1}$ while $0\leq x,y \leq 1$

$\frac1{(a+1)^2}+ \frac1{(b+1)^2}\geq\frac 1{1+ab}$ for positive reals $a$ and $b$

Deriving Cauchy-Schwarz inequality for random variables

Proving $\frac{a}{a^2+2b^2}+\frac{b}{b^2+2c^2}+\frac{c}{c^2+2a^2}\geq 1$ when $a^2+b^2+c^2=a^3+b^3+c^3$

Show that $\sqrt{\langle T(x), x \rangle}$ is a norm when $T$ is positive definite

Find the maximum of a sum

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