Question

$(a b+b c+a c)^2\left[\frac{1}{(a-b)^4}+\frac{1}{(b-c)^4}+\frac{1}{(c-a)^4}\right] \geq \frac{33}{16}$

score 70 · Answer 1 · 2023-07-27 12:56:28

70

Views

$(a b+b c+a c)^2\left[\frac{1}{(a-b)^4}+\frac{1}{(b-c)^4}+\frac{1}{(c-a)^4}\right] \geq \frac{33}{16}$

Published on 27 Jul 2023 - 12:56

score 71 · Answer 2 · 2023-07-30 18:26:28

71

Views

Proof of an Algebraic Identity

Published on 30 Jul 2023 - 18:26

score 110 · Answer 3 · 2026-03-26 04:34:41

110

Views

If $ab+bc+ca=1,$ prove $1+36(abc)^2\ge\frac{21abc}{a+b+c}. $

Published on 26 Mar 2026 - 4:34

score 71 · Answer 4 · 2026-03-26 02:56:23

71

Views

Prove $2(xy+yz+zx)+3\ge \sqrt{5x^2+4}+\sqrt{5y^2+4}+\sqrt{5z^2+4}$ when $x,y,z>0: x+y+z=3xyz.$

Published on 26 Mar 2026 - 2:56

score 44 · Answer 5 · 2023-08-03 13:40:19

44

Views

Can the elementary Symmetric polynomials be generated by powers of symmetric polynomials of degree 1?

Published on 03 Aug 2023 - 13:40

score 189 · Answer 6 · 2026-03-26 03:01:28

189

Views

If $abc=25,$ find maximum $M=\frac{1}{b+c+12a}+\frac{1}{c+a+12b}+\frac{1}{a+b+12c}.$

Published on 26 Mar 2026 - 3:01

score 63 · Answer 7 · 2026-03-26 04:53:04

63

Views

Find minimum and maximum $P=\frac{bc}{a^2+2b^2+2c^2}+\frac{ca}{b^2+2a^2+2c^2}+\frac{ab}{c^2+2b^2+2a^2}.$

Published on 26 Mar 2026 - 4:53

score 118 · Answer 8 · 2026-03-25 17:40:16

118

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If $a,b,c >0 : ab+bc+ca=3,$ find maximal value $\sum\dfrac{a\sqrt{a^2+2}}{a^2+3}$

Published on 25 Mar 2026 - 17:40

score 92 · Answer 9 · 2026-03-26 02:57:46

92

Views

Find the minimal constant $k$ :$\frac{a+k}{a+bc}+\frac{b+k}{b+ca}+\frac{c+k}{c+ab}\ge 3k+2,$ holds $ab+bc+ca=1.$

Published on 26 Mar 2026 - 2:57

score 120 · Answer 10 · 2026-03-26 02:57:41

120

Views

How to prove $\frac{5abc+1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1\right)+6\ge \sum_{\text{cyc}}\sqrt{5a^3+4}$ when $a^2+b^2+c^2=a+b+c$

Published on 26 Mar 2026 - 2:57

score 38 · Answer 11 · 2026-03-26 06:28:46

38

Views

Find minimum $\sum\frac{a^2(b+c)}{b^2+bc+c^2}$ when $a^2+b^2+c^2=a+b+c$

Published on 26 Mar 2026 - 6:28

score 74 · Answer 12 · 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 92 · Answer 13 · 2026-03-26 03:01:35

92

Views

Prove $\sum\dfrac{b+c}{a+bc}\ge 2\left[(a+b)(b+c)(c+a)+\frac{3abc}{a+b+c}\right]$ for $ab+bc+ca=1$

Published on 26 Mar 2026 - 3:01

score 144 · Answer 14 · 2026-03-26 02:57:52

144

Views

If $a+b+c=3,$ find max $T=\sqrt{\frac{6a+7bc}{6a+7}}+\sqrt{\frac{6b+7ca}{6b+7}}+\sqrt{\frac{6c+7ab}{6c+7}}$

Published on 26 Mar 2026 - 2:57

score 17 · Answer 15 · 2026-03-14 03:47:15

$p(x,y)=p(xy,\frac{1}{y}) (*)$ prove $\exists s(x,y),r(x,y)$ that apply to the Eq $(*)$ and $\exists q(x,y)$ so that $p(x,y)=q(s(x,y),r(x,y))$

List Question

$(a b+b c+a c)^2\left[\frac{1}{(a-b)^4}+\frac{1}{(b-c)^4}+\frac{1}{(c-a)^4}\right] \geq \frac{33}{16}$

Proof of an Algebraic Identity

If $ab+bc+ca=1,$ prove $1+36(abc)^2\ge\frac{21abc}{a+b+c}. $

Prove $2(xy+yz+zx)+3\ge \sqrt{5x^2+4}+\sqrt{5y^2+4}+\sqrt{5z^2+4}$ when $x,y,z>0: x+y+z=3xyz.$

Can the elementary Symmetric polynomials be generated by powers of symmetric polynomials of degree 1?

If $abc=25,$ find maximum $M=\frac{1}{b+c+12a}+\frac{1}{c+a+12b}+\frac{1}{a+b+12c}.$

Find minimum and maximum $P=\frac{bc}{a^2+2b^2+2c^2}+\frac{ca}{b^2+2a^2+2c^2}+\frac{ab}{c^2+2b^2+2a^2}.$

If $a,b,c >0 : ab+bc+ca=3,$ find maximal value $\sum\dfrac{a\sqrt{a^2+2}}{a^2+3}$

Find the minimal constant $k$ :$\frac{a+k}{a+bc}+\frac{b+k}{b+ca}+\frac{c+k}{c+ab}\ge 3k+2,$ holds $ab+bc+ca=1.$

How to prove $\frac{5abc+1}{4}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1\right)+6\ge \sum_{\text{cyc}}\sqrt{5a^3+4}$ when $a^2+b^2+c^2=a+b+c$

Find minimum $\sum\frac{a^2(b+c)}{b^2+bc+c^2}$ when $a^2+b^2+c^2=a+b+c$

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 $a+b+c=3$

Prove $\sum\dfrac{b+c}{a+bc}\ge 2\left[(a+b)(b+c)(c+a)+\frac{3abc}{a+b+c}\right]$ for $ab+bc+ca=1$

If $a+b+c=3,$ find max $T=\sqrt{\frac{6a+7bc}{6a+7}}+\sqrt{\frac{6b+7ca}{6b+7}}+\sqrt{\frac{6c+7ab}{6c+7}}$

$p(x,y)=p(xy,\frac{1}{y}) ()$ prove $\exists s(x,y),r(x,y)$ that apply to the Eq $()$ and $\exists q(x,y)$ so that $p(x,y)=q(s(x,y),r(x,y))$

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