Question

Using AM-GM inequality prove that $(1+\sqrt{2})(1+\sqrt{3})(1+\sqrt{5}) \lt 8\sqrt{30}$.

score 55 · Answer 1 · 2019-09-24 17:47:11

55

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Using AM-GM inequality prove that $(1+\sqrt{2})(1+\sqrt{3})(1+\sqrt{5}) \lt 8\sqrt{30}$.

Published on 24 Sep 2019 - 17:47

score 228 · Answer 2 · 2026-03-27 10:09:56

228

Views

Given three positive numbers $a,b,c$. Prove that $\sum\limits_{sym}\frac{a+b}{c}\geqq 2\sqrt{(\sum\limits_{sym}a)(\sum\limits_{sym}\frac{a}{bc}})$ .

Published on 27 Mar 2026 - 10:09

score 64 · Answer 3 · 2026-03-25 13:35:00

64

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Given three positive numbers $x,y,z$ so that $xyz=1, xy+yz+zx=5$. Find the maximum value $x^{c}+ y^{c}+ z^{c}$ for $c\geqq -1$ .

Published on 25 Mar 2026 - 13:35

score 92 · Answer 4 · 2019-09-30 03:50:47

92

Views

Prove Inequality $(a + b + c)d - (ab+ bc +ca) \leq 0 $ if $(ab + bc + ca)d - 3abc \leq 0$ .

Published on 30 Sep 2019 - 3:50

score 101 · Answer 5 · 2019-09-30 17:12:54

101

Views

Solving an inequality with three variables

Published on 30 Sep 2019 - 17:12

score 164 · Answer 6 · 2014-03-16 04:51:24

164

Views

How find the minimum of $ab+\frac{1}{a^2}+\frac{1}{b^2}$

Published on 16 Mar 2014 - 4:51

score 9.4k · Answer 7 · 2026-03-25 07:43:10

9.4k

Views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

Published on 25 Mar 2026 - 7:43

score 270 · Answer 8 · 2014-04-09 17:03:29

270

Views

AM-GM inequality

Published on 09 Apr 2014 - 17:03

score 174 · Answer 9 · 2026-03-25 03:22:13

174

Views

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Published on 25 Mar 2026 - 3:22

score 199 · Answer 10 · 2014-04-11 22:19:22

199

Views

Prove that positive $x,y$ satisfy $\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$.

Published on 11 Apr 2014 - 22:19

score 258 · Answer 11 · 2026-03-25 11:16:50

258

Views

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $(a+b+{1\over{ab}})$

Published on 25 Mar 2026 - 11:16

score 276 · Answer 12 · 2014-04-18 19:23:19

276

Views

Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$

Published on 18 Apr 2014 - 19:23

score 68 · Answer 13 · 2026-03-26 03:10:35

68

Views

Calculate the inequality

Published on 26 Mar 2026 - 3:10

score 153 · Answer 14 · 2014-05-04 21:51:07

153

Views

Prove or disprove inequality $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}$.

Published on 04 May 2014 - 21:51

score 626 · Answer 15 · 2026-03-25 12:49:48

626

Views

How to prove $\sqrt{\frac{ab}{2a^2+bc+ca}}+\sqrt{\frac{bc}{2b^2+ca+ab}}+\sqrt{\frac{ca}{2c^2+ab+bc}}\ge\frac{81}{2}\cdot\frac{abc}{(a+b+c)^3}$

Published on 25 Mar 2026 - 12:49

List Question

Using AM-GM inequality prove that $(1+\sqrt{2})(1+\sqrt{3})(1+\sqrt{5}) \lt 8\sqrt{30}$.

Given three positive numbers $a,b,c$. Prove that $\sum\limits_{sym}\frac{a+b}{c}\geqq 2\sqrt{(\sum\limits_{sym}a)(\sum\limits_{sym}\frac{a}{bc}})$ .

Given three positive numbers $x,y,z$ so that $xyz=1, xy+yz+zx=5$. Find the maximum value $x^{c}+ y^{c}+ z^{c}$ for $c\geqq -1$ .

Prove Inequality $(a + b + c)d - (ab+ bc +ca) \leq 0 $ if $(ab + bc + ca)d - 3abc \leq 0$ .

Solving an inequality with three variables

How find the minimum of $ab+\frac{1}{a^2}+\frac{1}{b^2}$

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

AM-GM inequality

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Prove that positive $x,y$ satisfy $\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$.

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $(a+b+{1\over{ab}})$

Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$

Calculate the inequality

Prove or disprove inequality $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}$.

How to prove $\sqrt{\frac{ab}{2a^2+bc+ca}}+\sqrt{\frac{bc}{2b^2+ca+ab}}+\sqrt{\frac{ca}{2c^2+ab+bc}}\ge\frac{81}{2}\cdot\frac{abc}{(a+b+c)^3}$

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