Question

Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

score 66 · Answer 1 · 2022-02-27 18:45:27

66

Views

Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

Published on 27 Feb 2022 - 18:45

score 78 · Answer 2 · 2022-02-28 17:22:01

78

Views

Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$

Published on 28 Feb 2022 - 17:22

score 84 · Answer 3 · 2022-03-10 20:38:13

84

Views

Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\frac ab} + \sqrt[5]{\frac bc} + \sqrt[7]{\frac ca} > \frac 52$

Published on 10 Mar 2022 - 20:38

score 202 · Answer 4 · 2022-03-11 04:02:39

202

Views

What is the minimum value of $\frac{2a}{a+2b+2c} + \frac{2b}{b+2c+2a} +\frac{2c}{c+2a+2b}$ given $ a+b+c=2$?

Published on 11 Mar 2022 - 4:02

score 324 · Answer 5 · 2022-03-12 14:41:13

324

Views

Prove that for $a,b,c \in \mathbb{R}^+$ with $abc = 1$, $\frac{1}{a^3(b + c)} + \frac{1}{b^3(a + c)} + \frac{1}{c^3(a + b)} \ge \frac{3}{2}$

Published on 12 Mar 2022 - 14:41

score 119 · Answer 6 · 2022-03-21 06:12:00

119

Views

Prove $\left(\frac{a+b+c}{3}\right)^p\leq \frac{a^p+b^p+c^p}{3}$.

Published on 21 Mar 2022 - 6:12

score 64 · Answer 7 · 2022-03-22 14:39:54

64

Views

Alternate proofs of the inequality $\sum_{\text{cyclic}} \frac{x}{2x+y+z}\le \frac{3}{4}$.

Published on 22 Mar 2022 - 14:39

score 77 · Answer 8 · 2022-03-23 11:37:01

77

Views

Is the geometric mean bounded above by this value?

Published on 23 Mar 2022 - 11:37

score 165 · Answer 9 · 2022-04-02 21:32:19

165

Views

Find minimum of the function using AM-GM

Published on 02 Apr 2022 - 21:32

score 67 · Answer 10 · 2022-04-04 10:21:22

67

Views

Can we prove the inequality without opening the parentheses? $(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$

Published on 04 Apr 2022 - 10:21

score 116 · Answer 11 · 2022-04-05 23:53:02

116

Views

Is this nice-looking inequality actually trivial?

Published on 05 Apr 2022 - 23:53

score 99 · Answer 12 · 2022-04-06 02:26:42

99

Views

Prove that $(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$

Published on 06 Apr 2022 - 2:26

score 135 · Answer 13 · 2022-04-09 17:00:02

135

Views

Showing $\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$ for $x,y,z>0$ and $xyz=1$

Published on 09 Apr 2022 - 17:00

score 80 · Answer 14 · 2022-04-11 09:36:33

80

Views

Prove $\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $

Published on 11 Apr 2022 - 9:36

score 168 · Answer 15 · 2022-04-14 07:03:09

168

Views

Prove the inequality $9(a+b)(b+c)(c+a) \geq 8(a+b+c)(ab+bc+ca)$

Published on 14 Apr 2022 - 7:03

List Question

Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$

Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\frac ab} + \sqrt[5]{\frac bc} + \sqrt[7]{\frac ca} > \frac 52$

What is the minimum value of $\frac{2a}{a+2b+2c} + \frac{2b}{b+2c+2a} +\frac{2c}{c+2a+2b}$ given $ a+b+c=2$?

Prove that for $a,b,c \in \mathbb{R}^+$ with $abc = 1$, $\frac{1}{a^3(b + c)} + \frac{1}{b^3(a + c)} + \frac{1}{c^3(a + b)} \ge \frac{3}{2}$

Prove $\left(\frac{a+b+c}{3}\right)^p\leq \frac{a^p+b^p+c^p}{3}$.

Alternate proofs of the inequality $\sum_{\text{cyclic}} \frac{x}{2x+y+z}\le \frac{3}{4}$.

Is the geometric mean bounded above by this value?

Find minimum of the function using AM-GM

Can we prove the inequality without opening the parentheses? $(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$

Is this nice-looking inequality actually trivial?

Prove that $(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$

Showing $\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$ for $x,y,z>0$ and $xyz=1$

Prove $\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $

Prove the inequality $9(a+b)(b+c)(c+a) \geq 8(a+b+c)(ab+bc+ca)$

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