Question

Show that $\sqrt[n]{x_1x_2\dots x_n}+\sqrt[n]{(1-x_1)(1-x_2)\dots(1-x_n)}\leq 1$

score 129 · Answer 1 · 2023-07-09 09:51:41

129

Views

Show that $\sqrt[n]{x_1x_2\dots x_n}+\sqrt[n]{(1-x_1)(1-x_2)\dots(1-x_n)}\leq 1$

Published on 09 Jul 2023 - 9:51

score 80 · Answer 2 · 2023-07-15 04:47:31

80

Views

Finding $\small{\min\limits_{a+b+c=3}\sqrt{ab+bc+ca}\left(\frac1{\sqrt{bc+a}}+\frac1{\sqrt{ca+b}}+\frac1{\sqrt{ab+c}}\right).}$

Published on 15 Jul 2023 - 4:47

score 208 · Answer 3 · 2023-07-16 07:22:29

208

Views

$a+b+c+abc=4,$ prove $\sum_{cyc}\frac{1}{\sqrt{a^2+b^2}}\ge \frac{4+\sqrt{2}}{4}.$

Published on 16 Jul 2023 - 7:22

score 82 · Answer 4 · 2026-03-26 01:23:23

82

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a,b,c>0 and prove $\sum\frac{\sqrt{a+b}}{c+\sqrt{ab}}\ge \frac{3\sqrt{2}}{2}\sqrt{\frac{a+b+c}{ab+bc+ca}}.$

Published on 26 Mar 2026 - 1:23

score 145 · Answer 5 · 2023-07-19 22:05:47

145

Views

Find the maximum of the expression, given that $a+b+c+d=100$

Published on 19 Jul 2023 - 22:05

score 232 · Answer 6 · 2026-03-25 22:05:10

232

Views

Prove $\frac{1}{\sqrt{a+8b}}+\frac{1}{\sqrt{b+8c}}+\frac{1}{\sqrt{c+8a}}\ge 1$ for $ab + bc + ca = 3$

Published on 25 Mar 2026 - 22:05

score 122 · Answer 7 · 2023-07-23 23:23:53

122

Views

Find the image of function $f$ (without calculus)

Published on 23 Jul 2023 - 23:23

score 141 · Answer 8 · 2026-03-25 22:05:02

141

Views

If $ab+bc+ca=3$, then prove that $\sum\sqrt{\frac{b+c}{bc+1}}\ge\frac{a+b+c+3}{\sqrt{a+b+c+abc}}$

Published on 25 Mar 2026 - 22:05

score 99 · Answer 9 · 2023-07-24 08:58:42

99

Views

Holder inequality for multiple sums?

Published on 24 Jul 2023 - 8:58

score 89 · Answer 10 · 2023-07-25 09:36:33

Prove: $\left(\sum_{i=1}^na_ix_i^2\right)\left(\sum_{i=1}^n a_iy_i\right)\geq\left(\sum_{i=1}^n a_ix_i\right)\left(\sum_{i=1}^n a_ix_iy_i\right)$

score 82 · Answer 11 · 2023-07-27 04:24:14

82

Views

An application of Holder's inequality involving the infinity norm

Published on 27 Jul 2023 - 4:24

score 184 · Answer 12 · 2023-07-27 11:32:18

184

Views

$a,b,c>0:a+b+c=3.$ Prove that: $\sum\sqrt{\frac{ab+2}{ab+c}}\ge \frac{3\sqrt{6}}{2}.$

Published on 27 Jul 2023 - 11:32

score 28 · Answer 13 · 2026-03-25 23:38:31

28

Views

The Inner Product of a Hadamard Product

Published on 25 Mar 2026 - 23:38

score 537 · Answer 14 · 2023-07-28 05:04:20

537

Views

Prove $\sum\limits_{\mathrm{cyc}} \sqrt{5a+5b+8ab}\ge 3\sqrt{2}+2\sqrt{5}$ for $ab+bc+ca=1$

Published on 28 Jul 2023 - 5:04

score 204 · Answer 15 · 2023-07-29 01:07:17

204

Views

A proof to Holder Inequality

Published on 29 Jul 2023 - 1:07

List Question

Show that $\sqrt[n]{x_1x_2\dots x_n}+\sqrt[n]{(1-x_1)(1-x_2)\dots(1-x_n)}\leq 1$

Finding $\small{\min\limits_{a+b+c=3}\sqrt{ab+bc+ca}\left(\frac1{\sqrt{bc+a}}+\frac1{\sqrt{ca+b}}+\frac1{\sqrt{ab+c}}\right).}$

$a+b+c+abc=4,$ prove $\sum_{cyc}\frac{1}{\sqrt{a^2+b^2}}\ge \frac{4+\sqrt{2}}{4}.$

a,b,c>0 and prove $\sum\frac{\sqrt{a+b}}{c+\sqrt{ab}}\ge \frac{3\sqrt{2}}{2}\sqrt{\frac{a+b+c}{ab+bc+ca}}.$

Find the maximum of the expression, given that $a+b+c+d=100$

Prove $\frac{1}{\sqrt{a+8b}}+\frac{1}{\sqrt{b+8c}}+\frac{1}{\sqrt{c+8a}}\ge 1$ for $ab + bc + ca = 3$

Find the image of function $f$ (without calculus)

If $ab+bc+ca=3$, then prove that $\sum\sqrt{\frac{b+c}{bc+1}}\ge\frac{a+b+c+3}{\sqrt{a+b+c+abc}}$

Holder inequality for multiple sums?

Prove: $\left(\sum_{i=1}^na_ix_i^2\right)\left(\sum_{i=1}^n a_iy_i\right)\geq\left(\sum_{i=1}^n a_ix_i\right)\left(\sum_{i=1}^n a_ix_iy_i\right)$

An application of Holder's inequality involving the infinity norm

$a,b,c>0:a+b+c=3.$ Prove that: $\sum\sqrt{\frac{ab+2}{ab+c}}\ge \frac{3\sqrt{6}}{2}.$

The Inner Product of a Hadamard Product

Prove $\sum\limits_{\mathrm{cyc}} \sqrt{5a+5b+8ab}\ge 3\sqrt{2}+2\sqrt{5}$ for $ab+bc+ca=1$

A proof to Holder Inequality

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