Question

If $a+b+c=0$ prove that $ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $

score 227 · Answer 1 · 2017-10-12 20:55:21

227

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If $a+b+c=0$ prove that $ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $

Published on 12 Oct 2017 - 20:55

score 186 · Answer 2 · 2017-11-15 06:52:32

186

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maximum and minimum of $\frac{a^4+b^4+c^4}{(a+b+c)^4},$

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283

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score 119 · Answer 4 · 2026-03-25 17:38:55

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score 1.1k · Answer 6 · 2012-08-31 05:17:31

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Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

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score 7.2k · Answer 7 · 2012-09-17 10:00:36

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How to prove this inequality $\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}}$

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score 348 · Answer 8 · 2026-03-22 01:38:19

348

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Showing that: $(\frac{a}{b+c})^2+(\frac{b}{a+c})^2+(\frac{c}{a+b})^2+\frac{10abc}{(a+b)(b+c)(c+a)}\ge 2$

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score 5.6k · Answer 9 · 2013-03-03 13:05:42

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Prove that $\frac 1 {x+y}+\frac 1 {y+z}+\frac 1 {z+x}\geq \frac 5 2$.

Published on 03 Mar 2013 - 13:05

score 3.5k · Answer 10 · 2013-03-15 07:05:38

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How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$

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score 257 · Answer 11 · 2026-03-25 15:47:10

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score 101 · Answer 12 · 2026-03-25 19:10:19

101

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Proving $\displaystyle \sum_{cyc}\frac{(a^2+b^2)}{a+b}\leqslant \frac{3(a^2+b^2+c^2)}{a+b+c}$

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score 113 · Answer 13 · 2026-03-22 14:28:34

113

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Proving $ \sum \frac{a(b+c)}{a^2+bc}+\frac{2\sum (a^2-bc)}{(a+b+c)^2}+\frac{96(a-b)^2(b-c)^2(c-a)^2}{(a+b+c)^6} \leqslant \frac{(a+b+c)^2}{ab+bc+ca} $

Published on 22 Mar 2026 - 14:28

score 115 · Answer 14 · 2020-08-20 02:05:48

115

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Find the stronger version of $9 ( a+b+c ) ^{2} ( ab+ac+bc ) ^{2}+108a^2b^2c^2-31abc ( a+b+c ) ^{3} \geqslant 0$

Published on 20 Aug 2020 - 2:05

score 133 · Answer 15 · 2026-03-25 19:10:26

133

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Proving $\frac{a^3+b^3+c^3}{3}-abc\ge \frac{3}{4}\sqrt{(a-b)^2(b-c)^2(c-a)^2}$

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List Question

If $a+b+c=0$ prove that $ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $

maximum and minimum of $\frac{a^4+b^4+c^4}{(a+b+c)^4},$

Finding the maximum of $p^3 + q^3 +r^3 + 4pqr$

How to solve (in)equality of three variables with trigonometric solutions

Combined AM GM QM inequality

Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

How to prove this inequality $\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}}$

Showing that: $(\frac{a}{b+c})^2+(\frac{b}{a+c})^2+(\frac{c}{a+b})^2+\frac{10abc}{(a+b)(b+c)(c+a)}\ge 2$

Prove that $\frac 1 {x+y}+\frac 1 {y+z}+\frac 1 {z+x}\geq \frac 5 2$.

How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$

Proving $(1+a^2)(1+b^2)(1+c^2)\geq8 $

Proving $\displaystyle \sum_{cyc}\frac{(a^2+b^2)}{a+b}\leqslant \frac{3(a^2+b^2+c^2)}{a+b+c}$

Proving $ \sum \frac{a(b+c)}{a^2+bc}+\frac{2\sum (a^2-bc)}{(a+b+c)^2}+\frac{96(a-b)^2(b-c)^2(c-a)^2}{(a+b+c)^6} \leqslant \frac{(a+b+c)^2}{ab+bc+ca} $

Find the stronger version of $9 ( a+b+c ) ^{2} ( ab+ac+bc ) ^{2}+108a^2b^2c^2-31abc ( a+b+c ) ^{3} \geqslant 0$

Proving $\frac{a^3+b^3+c^3}{3}-abc\ge \frac{3}{4}\sqrt{(a-b)^2(b-c)^2(c-a)^2}$

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