Question

For $abc=1$ prove that $\sum\limits_\text{cyc}\frac{1}{a+3}\geq\sum\limits_\text{cyc}\frac{a}{a^2+3}$

score 698 · Answer 1 · 2017-01-09 19:53:01

698

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For $abc=1$ prove that $\sum\limits_\text{cyc}\frac{1}{a+3}\geq\sum\limits_\text{cyc}\frac{a}{a^2+3}$

Published on 09 Jan 2017 - 19:53

score 739 · Answer 2 · 2026-03-25 09:29:06

739

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Proving $\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$

Published on 25 Mar 2026 - 9:29

score 1.9k · Answer 3 · 2017-01-30 12:44:03

1.9k

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Why is the "Buffalo Way" considered inelegant?

Published on 30 Jan 2017 - 12:44

score 111 · Answer 4 · 2017-03-15 21:28:38

111

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For what values of $k>0$ does $abc=1 \implies \sum_{\mbox{cyc}}\left(\frac1{a+k}-\frac{a}{a^2+k}\right) \geq 0$?

Published on 15 Mar 2017 - 21:28

score 280 · Answer 5 · 2017-03-24 10:21:27

280

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How find this maximum of the value $\sum_{i=1}^{6}x_{i}x_{i+1}x_{i+2}x_{i+3}$?

Published on 24 Mar 2017 - 10:21

score 341 · Answer 6 · 2017-03-31 06:41:17

341

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Prove that $3(a^5b+b^5c+c^5a)\geq(a^2c+b^2a+c^2b)^2$

Published on 31 Mar 2017 - 6:41

score 241 · Answer 7 · 2017-04-02 14:40:50

241

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If $a+b+c=ab+ac+bc$ then $\sum\limits_{cyc}\frac{a}{a^2+2b}\leq1$.

Published on 02 Apr 2017 - 14:40

score 251 · Answer 8 · 2023-07-31 10:46:02

251

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If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$

Published on 31 Jul 2023 - 10:46

score 91 · Answer 9 · 2023-08-04 07:01:25

91

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Prove $\sum\frac{a}{(2k^2-3k+2)a+k^2b+c}\le \frac{1}{k^2-k+1}$

Published on 04 Aug 2023 - 7:01

score 49 · Answer 10 · 2023-08-07 01:49:14

49

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How to find minimum $T= a+b+(a-b)(c-b)$ for $ab+bc+ca=3.$

Published on 07 Aug 2023 - 1:49

score 141 · Answer 11 · 2023-08-08 14:51:16

141

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Finding $\small{\max\limits_{a+b+c=3}\frac{1}{\sqrt{a^2-3a+3}}+\frac{1}{\sqrt{b^2-3b+3}}+\frac{1}{\sqrt{c^2-3c+3}}.}$

Published on 08 Aug 2023 - 14:51

score 85 · Answer 12 · 2023-08-10 05:27:21

85

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Help to prove $\frac{1}{4a^2-8ac+13c^2}+\frac{1}{4b^2-8bc+13c^2}\ge \frac{2}{(a+b+c)^2}$

Published on 10 Aug 2023 - 5:27

score 117 · Answer 13 · 2023-11-23 14:42:44

117

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Prove $\sqrt{2a+b^3}+\sqrt{2b+c^3}+\sqrt{2c+a^3}\le 3\sqrt{3}$ for $a^2+b^2+c^2=3.$

Published on 23 Nov 2023 - 14:42

score 96 · Answer 14 · 2023-11-24 14:34:13

96

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If $a^2+b^2+c^2=3,$ then prove $\sqrt{\frac{a+b}{4a+3}}+\sqrt{\frac{b+c}{4b+3}}+\sqrt{\frac{c+a}{4c+3}}\le 3\sqrt{\frac{2}{7}}.$

Published on 24 Nov 2023 - 14:34

score 139 · Answer 15 · 2023-11-25 12:16:00

139

Views

Finding $\small{\min\limits_{a+b+c=2}\sqrt{a-2bc+3}+\sqrt{b-2ca+3}+\sqrt{c-2ab+3}.}$

Published on 25 Nov 2023 - 12:16

List Question

For $abc=1$ prove that $\sum\limits_\text{cyc}\frac{1}{a+3}\geq\sum\limits_\text{cyc}\frac{a}{a^2+3}$

Proving $\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$

Why is the "Buffalo Way" considered inelegant?

For what values of $k>0$ does $abc=1 \implies \sum_{\mbox{cyc}}\left(\frac1{a+k}-\frac{a}{a^2+k}\right) \geq 0$?

How find this maximum of the value $\sum_{i=1}^{6}x_{i}x_{i+1}x_{i+2}x_{i+3}$?

Prove that $3(a^5b+b^5c+c^5a)\geq(a^2c+b^2a+c^2b)^2$

If $a+b+c=ab+ac+bc$ then $\sum\limits_{cyc}\frac{a}{a^2+2b}\leq1$.

If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$

Prove $\sum\frac{a}{(2k^2-3k+2)a+k^2b+c}\le \frac{1}{k^2-k+1}$

How to find minimum $T= a+b+(a-b)(c-b)$ for $ab+bc+ca=3.$

Finding $\small{\max\limits_{a+b+c=3}\frac{1}{\sqrt{a^2-3a+3}}+\frac{1}{\sqrt{b^2-3b+3}}+\frac{1}{\sqrt{c^2-3c+3}}.}$

Help to prove $\frac{1}{4a^2-8ac+13c^2}+\frac{1}{4b^2-8bc+13c^2}\ge \frac{2}{(a+b+c)^2}$

Prove $\sqrt{2a+b^3}+\sqrt{2b+c^3}+\sqrt{2c+a^3}\le 3\sqrt{3}$ for $a^2+b^2+c^2=3.$

If $a^2+b^2+c^2=3,$ then prove $\sqrt{\frac{a+b}{4a+3}}+\sqrt{\frac{b+c}{4b+3}}+\sqrt{\frac{c+a}{4c+3}}\le 3\sqrt{\frac{2}{7}}.$

Finding $\small{\min\limits_{a+b+c=2}\sqrt{a-2bc+3}+\sqrt{b-2ca+3}+\sqrt{c-2ab+3}.}$

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