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 134 · Answer 1 · 2026-04-17 08:16:36

134

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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 17 Apr 2026 - 8:16

score 255 · Answer 2 · 2026-03-25 12:28:15

255

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 25 Mar 2026 - 12:28

score 91 · Answer 3 · 2026-03-25 12:26:14

91

Views

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

Published on 25 Mar 2026 - 12:26

score 74 · Answer 4 · 2026-03-25 12:28:17

74

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 25 Mar 2026 - 12:28

score 75 · Answer 5 · 2026-03-25 12:27:22

75

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 25 Mar 2026 - 12:27

score 170 · Answer 6 · 2026-03-25 12:27:21

170

Views

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

Published on 25 Mar 2026 - 12:27

score 325 · Answer 7 · 2026-03-25 12:28:22

325

Views

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

Published on 25 Mar 2026 - 12:28

score 271 · Answer 8 · 2026-05-01 07:15:35

271

Views

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

Published on 01 May 2026 - 7:15

score 196 · Answer 9 · 2026-04-24 23:14:07

196

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 24 Apr 2026 - 23:14

score 114 · Answer 10 · 2026-03-25 12:29:22

114

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 25 Mar 2026 - 12:29

score 1.6k · Answer 11 · 2026-04-16 22:02:24

1.6k

Views

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

Published on 16 Apr 2026 - 22:02

score 246 · Answer 12 · 2026-03-25 12:27:07

246

Views

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

Published on 25 Mar 2026 - 12:27

score 72 · Answer 13 · 2026-03-25 12:26:02

72

Views

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

Published on 25 Mar 2026 - 12:26

score 98 · Answer 14 · 2026-03-25 12:27:21

98

Views

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

Published on 25 Mar 2026 - 12:27

score 115 · Answer 15 · 2026-03-25 12:27:25

115

Views

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

Published on 25 Mar 2026 - 12:27

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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