Question

If $27 + abc = (a +b +c)(ab + ac+ bc)$ so $\frac{a^2}{a+2b} + \frac{b^2}{b+2c} + \frac{c^2}{c+2a }\geq \frac{3}{2}$

score 115 · Answer 1 · 2026-04-17 06:48:08

115

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If $27 + abc = (a +b +c)(ab + ac+ bc)$ so $\frac{a^2}{a+2b} + \frac{b^2}{b+2c} + \frac{c^2}{c+2a }\geq \frac{3}{2}$

Published on 17 Apr 2026 - 6:48

score 181 · Answer 2 · 2026-03-26 01:16:46

181

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How prove this converse cauchy inequality of problem?

Published on 26 Mar 2026 - 1:16

score 171 · Answer 3 · 2026-04-14 17:37:01

171

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Prove the inequality $\sum\limits_{cyc}\sqrt{x+yz}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}$

Published on 14 Apr 2026 - 17:37

score 151 · Answer 4 · 2026-03-25 21:51:08

151

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If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Published on 25 Mar 2026 - 21:51

score 662 · Answer 5 · 2026-03-26 01:17:55

662

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Given $ab+bc+ca=3abc$, prove $\sqrt{\frac{a+b}{c(a^2+b^2 )}}+\sqrt{\frac{b+c}{a(b^2+c^2)}}+\sqrt{\frac{c+a}{b(c^2+a^2 )}}\leq 3$

Published on 26 Mar 2026 - 1:17

score 115 · Answer 6 · 2026-04-14 16:33:36

115

Views

Proving $\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$

Published on 14 Apr 2026 - 16:33

score 275 · Answer 7 · 2026-04-15 19:49:52

275

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AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$

Published on 15 Apr 2026 - 19:49

score 80 · Answer 8 · 2026-04-14 12:42:06

80

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How prove $\sum\limits_{cyc}\sqrt{a^2+b^2-c^2}\sqrt{a^2-b^2+c^2}\le ab+bc+ca$

Published on 14 Apr 2026 - 12:42

score 296 · Answer 9 · 2026-03-26 01:16:41

296

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If $a+b+c = 6$ and $a$, $b$, $c$ are nonnegative then $a^2+b^2+c^2 \geq 12$

Published on 26 Mar 2026 - 1:16

score 232 · Answer 10 · 2026-04-15 22:28:27

232

Views

Prove $\frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ac}{1+a+c}\geq \frac{3}{2}$ for $a$, $b$, $c$ positive and $1+a+b+c=2abc$

Published on 15 Apr 2026 - 22:28

score 135 · Answer 11 · 2026-04-11 04:09:09

135

Views

Difficult trigonometry problem to find the minimum value

Published on 11 Apr 2026 - 4:09

score 122 · Answer 12 · 2026-04-11 10:08:23

122

Views

maximum value of expression $6bc+6abc+2ab+3ac$

Published on 11 Apr 2026 - 10:08

score 270 · Answer 13 · 2026-04-13 23:02:10

270

Views

How to show this inequality $\sum\sqrt{\frac{x}{x+2y+z}}\le 2$

Published on 13 Apr 2026 - 23:02

score 21.1k · Answer 14 · 2026-04-16 11:05:54

21.1k

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Finding maxima and minima of $f(x,y)= x^4 + y^4 - 2x^2 - 2y^2 + 4xy$

Published on 16 Apr 2026 - 11:05

score 185 · Answer 15 · 2026-04-14 15:29:56

185

Views

Proving an inequality via the Cauchy-Schwarz Inequality

Published on 14 Apr 2026 - 15:29

List Question

If $27 + abc = (a +b +c)(ab + ac+ bc)$ so $\frac{a^2}{a+2b} + \frac{b^2}{b+2c} + \frac{c^2}{c+2a }\geq \frac{3}{2}$

How prove this converse cauchy inequality of problem?

Prove the inequality $\sum\limits_{cyc}\sqrt{x+yz}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}$

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Given $ab+bc+ca=3abc$, prove $\sqrt{\frac{a+b}{c(a^2+b^2 )}}+\sqrt{\frac{b+c}{a(b^2+c^2)}}+\sqrt{\frac{c+a}{b(c^2+a^2 )}}\leq 3$

Proving $\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$

AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$

How prove $\sum\limits_{cyc}\sqrt{a^2+b^2-c^2}\sqrt{a^2-b^2+c^2}\le ab+bc+ca$

If $a+b+c = 6$ and $a$, $b$, $c$ are nonnegative then $a^2+b^2+c^2 \geq 12$

Prove $\frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ac}{1+a+c}\geq \frac{3}{2}$ for $a$, $b$, $c$ positive and $1+a+b+c=2abc$

Difficult trigonometry problem to find the minimum value

maximum value of expression $6bc+6abc+2ab+3ac$

How to show this inequality $\sum\sqrt{\frac{x}{x+2y+z}}\le 2$

Finding maxima and minima of $f(x,y)= x^4 + y^4 - 2x^2 - 2y^2 + 4xy$

Proving an inequality via the Cauchy-Schwarz Inequality

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