Question

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

score 349 · Answer 1 · 2026-03-26 14:23:06

349

Views

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

Published on 26 Mar 2026 - 14:23

score 184 · Answer 2 · 2026-03-22 14:10:07

184

Views

Help with inequality please

Published on 22 Mar 2026 - 14:10

score 124 · Answer 3 · 2026-03-21 23:41:26

124

Views

Calculate list of all pairs $(x,y)$

Published on 21 Mar 2026 - 23:41

score 158 · Answer 4 · 2026-03-22 00:15:47

158

Views

On proving $a^3+b^3+c^3-3abc \geq 2\left({b+c\over 2}-a\right)^3$.

Published on 22 Mar 2026 - 0:15

score 114 · Answer 5 · 2026-03-26 14:22:51

114

Views

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 26 Mar 2026 - 14:22

score 295 · Answer 6 · 2026-03-25 19:05:10

295

Views

Proving $3(a^4 + b^4 + c^4 + d^4) + 4abcd \geq (a+b+c+d)(a^3 + b^3 + c^3 + d^3)$

Published on 25 Mar 2026 - 19:05

score 122 · Answer 7 · 2026-03-21 16:17:55

122

Views

Proving $\sum {\frac {ab}{ \left( a+b \right) ^{2}}}+{\frac {\prod \left( a+b \right) }{16abc}}\geq \frac{5}{4}$

Published on 21 Mar 2026 - 16:17

score 139 · Answer 8 · 2026-03-22 10:53:32

139

Views

Find the smallest possible value of an equation, where $a+b+c=3$

Published on 22 Mar 2026 - 10:53

score 182 · Answer 9 · 2026-03-22 04:38:11

182

Views

prove that $xy+yz+zx\ge x\sqrt{yz}+y\sqrt{xz}+z\sqrt{xy}$

Published on 22 Mar 2026 - 4:38

score 96 · Answer 10 · 2026-03-25 21:49:47

96

Views

prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

Published on 25 Mar 2026 - 21:49

score 137 · Answer 11 · 2026-03-21 20:34:45

137

Views

prove that $\sum_{cyc}\frac{{a^2}{b}}{c}\ge a^2+b^2+c^2$

Published on 21 Mar 2026 - 20:34

score 144 · Answer 12 · 2026-03-25 19:04:47

144

Views

Prove that $\frac{1}{abc}+36\ge \frac{21}{ab+bc+ca}$

Published on 25 Mar 2026 - 19:04

score 161 · Answer 13 · 2026-03-26 16:07:09

161

Views

Proving $\sum \frac{b+c}{9(a^2+3bc)+4(a+b+c)^2}\geqslant \frac{1}{4(a+b+c)}$

Published on 26 Mar 2026 - 16:07

score 197 · Answer 14 · 2026-03-25 17:44:34

197

Views

Proving ${\frac {35{x}^{2}+7x(y+z)+23yz}{35(x^2+y^2+z^2)+37(xy+yz+zx)}}\leqslant \sqrt {{\frac {{x}^{2}+yz}{6\,{y}^{2} +6\,yz+6\,{z}^{2}}}}$

Published on 25 Mar 2026 - 17:44

score 124 · Answer 15 · 2026-03-26 16:10:14

124

Views

Proving $Q=\frac{a^{3}+b^{3}+c^{3}}{(a+b)(b+c)(c+a)}+k\cdot \frac{(ab+bc+ca)}{(a+b+c)^{2}}\geqslant \frac{3}{8}+\frac{k}{3}$

Published on 26 Mar 2026 - 16:10

List Question

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

Help with inequality please

Calculate list of all pairs $(x,y)$

On proving $a^3+b^3+c^3-3abc \geq 2\left({b+c\over 2}-a\right)^3$.

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} $

Proving $3(a^4 + b^4 + c^4 + d^4) + 4abcd \geq (a+b+c+d)(a^3 + b^3 + c^3 + d^3)$

Proving $\sum {\frac {ab}{ \left( a+b \right) ^{2}}}+{\frac {\prod \left( a+b \right) }{16abc}}\geq \frac{5}{4}$

Find the smallest possible value of an equation, where $a+b+c=3$

prove that $xy+yz+zx\ge x\sqrt{yz}+y\sqrt{xz}+z\sqrt{xy}$

prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

prove that $\sum_{cyc}\frac{{a^2}{b}}{c}\ge a^2+b^2+c^2$

Prove that $\frac{1}{abc}+36\ge \frac{21}{ab+bc+ca}$

Proving $\sum \frac{b+c}{9(a^2+3bc)+4(a+b+c)^2}\geqslant \frac{1}{4(a+b+c)}$

Proving ${\frac {35{x}^{2}+7x(y+z)+23yz}{35(x^2+y^2+z^2)+37(xy+yz+zx)}}\leqslant \sqrt {{\frac {{x}^{2}+yz}{6\,{y}^{2} +6\,yz+6\,{z}^{2}}}}$

Proving $Q=\frac{a^{3}+b^{3}+c^{3}}{(a+b)(b+c)(c+a)}+k\cdot \frac{(ab+bc+ca)}{(a+b+c)^{2}}\geqslant \frac{3}{8}+\frac{k}{3}$

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