Question

Find the maximum value $\frac{11a+4b}{4a^2-ab+2b^2}+\frac{11b+4c}{4b^2-bc+2c^2}+\frac{11c+4a}{4c^2-ca+2a^2}$

score 131 · Answer 1 · 2017-04-05 15:20:42

131

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Find the maximum value $\frac{11a+4b}{4a^2-ab+2b^2}+\frac{11b+4c}{4b^2-bc+2c^2}+\frac{11c+4a}{4c^2-ca+2a^2}$

Published on 05 Apr 2017 - 15:20

score 254 · Answer 2 · 2023-02-10 09:28:43

254

Views

Inequality $\frac{a^3}{3ab^2+2c^3} +\frac{b^3}{3bc^2+2a^3} +\frac{c^3}{3ca^2+2b^3} \geq \frac{3}{5} $

Published on 10 Feb 2023 - 9:28

score 90 · Answer 3 · 2023-03-11 02:22:34

90

Views

Prove $\sum_{\mathrm{cyc}}\sqrt{\frac{a^3+a^2+1}{a^2+a+1}}\ge3$ for $abc=1$

Published on 11 Mar 2023 - 2:22

score 73 · Answer 4 · 2023-08-05 17:03:04

73

Views

How to prove $\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$ when $a+b+c=3$

Published on 05 Aug 2023 - 17:03

score 74 · Answer 5 · 2023-10-24 15:02:54

74

Views

Max $P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$

Published on 24 Oct 2023 - 15:02

score 169 · Answer 6 · 2023-11-21 20:53:45

169

Views

Inequality $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$

Published on 21 Nov 2023 - 20:53

score 324 · Answer 7 · 2015-01-08 00:27:20

324

Views

Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$

Published on 08 Jan 2015 - 0:27

score 269 · Answer 8 · 2026-03-18 07:02:24

269

Views

$ x+y+z = 3, \; \sum\limits_{cyc} \frac{x}{2x^2+x+1} \leq \frac{3}{4} $

Published on 18 Mar 2026 - 7:02

score 194 · Answer 9 · 2016-01-08 11:13:03

194

Views

Prove that $\frac{bc}{a^2+bc}+\frac{ca}{b^2+ca}+\frac{ab}{c^2+ab}\le{\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}}$

Published on 08 Jan 2016 - 11:13

score 113 · Answer 10 · 2013-07-12 15:54:17

113

Views

If $x,y,z>0$ are distinct and $x+y+z=1$ what is the minimum of $\left((1+x)(1+y)(1+z)\right)/\left((1-x)(1-y)(1-z)\right)$?

Published on 12 Jul 2013 - 15:54

score 1.6k · Answer 11 · 2014-03-05 20:24:04

1.6k

Views

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

Published on 05 Mar 2014 - 20:24

score 245 · Answer 12 · 2017-05-02 18:47:40

245

Views

$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$

Published on 02 May 2017 - 18:47

score 71 · Answer 13 · 2017-05-14 13:50:26

71

Views

Find maximize $A=\sum_{cyc}\sqrt{1+x^2}+2(\sum_{cyc}\sqrt{x})$

Published on 14 May 2017 - 13:50

score 97 · Answer 14 · 2017-05-15 15:57:25

97

Views

Prove that $\sum_{cyc}\frac{x}{y^2+z^2}\ge\frac{3\sqrt3}2$

Published on 15 May 2017 - 15:57

score 114 · Answer 15 · 2017-06-26 16:39:09

114

Views

How prove this inequality $\sum_{cyc}\sqrt{\frac{2b+2c}{a}-1}\ge 3\sqrt{3}$

Published on 26 Jun 2017 - 16:39

List Question

Find the maximum value $\frac{11a+4b}{4a^2-ab+2b^2}+\frac{11b+4c}{4b^2-bc+2c^2}+\frac{11c+4a}{4c^2-ca+2a^2}$

Inequality $\frac{a^3}{3ab^2+2c^3} +\frac{b^3}{3bc^2+2a^3} +\frac{c^3}{3ca^2+2b^3} \geq \frac{3}{5} $

Prove $\sum_{\mathrm{cyc}}\sqrt{\frac{a^3+a^2+1}{a^2+a+1}}\ge3$ for $abc=1$

How to prove $\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$ when $a+b+c=3$

Max $P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$

Inequality $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$

Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$

$ x+y+z = 3, \; \sum\limits_{cyc} \frac{x}{2x^2+x+1} \leq \frac{3}{4} $

Prove that $\frac{bc}{a^2+bc}+\frac{ca}{b^2+ca}+\frac{ab}{c^2+ab}\le{\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}}$

If $x,y,z>0$ are distinct and $x+y+z=1$ what is the minimum of $\left((1+x)(1+y)(1+z)\right)/\left((1-x)(1-y)(1-z)\right)$?

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$

Find maximize $A=\sum_{cyc}\sqrt{1+x^2}+2(\sum_{cyc}\sqrt{x})$

Prove that $\sum_{cyc}\frac{x}{y^2+z^2}\ge\frac{3\sqrt3}2$

How prove this inequality $\sum_{cyc}\sqrt{\frac{2b+2c}{a}-1}\ge 3\sqrt{3}$

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