Question

$(a^2+b^2+c^2+k(ab+bc+ca))(\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2})\geq \frac{9}{4}(2-k)$

score 42 · Answer 1 · 2026-03-26 02:57:56

42

Views

$(a^2+b^2+c^2+k(ab+bc+ca))(\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2})\geq \frac{9}{4}(2-k)$

Published on 26 Mar 2026 - 2:57

score 74 · Answer 2 · 2026-03-28 05:23:17

74

Views

How to prove $\frac{\sqrt{ab+bc+ca}}{2}\ge\frac{a^3(b+c)+b^3(c+a)+c^3(a+b)}{(a^2+b^2+c^2)(a+b+c)-2abc}.$?

Published on 28 Mar 2026 - 5:23

score 141 · Answer 3 · 2026-03-26 02:56:23

141

Views

If $x^2+y^2+z^2=1,$ prove $\frac{x(z^2-y^2+2y)}{z}+\frac{y(x^2-z^2+2z)}{x}+\frac{z(y^2-x^2+2x)}{y}\ge 2$

Published on 26 Mar 2026 - 2:56

score 187 · Answer 4 · 2026-03-25 15:42:13

187

Views

Finding $\small{\max\limits_{ab+bc+ca=3}\frac{ab}{2a+b}+\frac{bc}{2b+c}+\frac{ca}{2c+a}.}$

Published on 25 Mar 2026 - 15:42

score 58 · Answer 5 · 2026-03-26 06:34:49

58

Views

Find minimum $\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}$ when $a+b+c=4$

Published on 26 Mar 2026 - 6:34

score 119 · Answer 6 · 2026-03-26 02:55:57

119

Views

Finding $\small{\min\limits_{ab+bc+ca=3}\dfrac{1}{4a+bc}+\dfrac{1}{4b+ca}+\dfrac{1}{4c+ab}.}$

Published on 26 Mar 2026 - 2:55

score 193 · Answer 7 · 2026-03-29 02:14:22

193

Views

A difficult minimum

Published on 29 Mar 2026 - 2:14

score 467 · Answer 8 · 2026-03-26 04:35:02

467

Views

Let $a+b+c=3,$ then prove$\sqrt[3]{\frac{a+b}{5ab+4}}+\sqrt[3]{\frac{c+b}{5cb+4}}+\sqrt[3]{\frac{a+c}{5ac+4}}\ge \sqrt[3]{6}$

Published on 26 Mar 2026 - 4:35

score 70 · Answer 9 · 2026-03-26 04:36:48

70

Views

Finding $\small{\max\limits_{ab+bc+ca=abc+2}\frac{ab(2-c)}{a^2+abc+b^2}+\frac{bc(2-a)}{b^2+abc+c^2}+\frac{ca(2-b)}{c^2+abc+a^2}.}$

Published on 26 Mar 2026 - 4:36

score 115 · Answer 10 · 2026-02-23 08:05:04

115

Views

For $x+y+z=3,$ prove $\frac{1}{x^2+4}+\frac{1}{y^2+4}+\frac{1}{z^2+4}\le \frac{3}{5}.$

Published on 23 Feb 2026 - 8:05

score 108 · Answer 11 · 2026-03-25 18:50:37

108

Views

If $x,y,z>0: x+y+z=3,$ prove $\frac{x^2}{y^2+yz+z^2}+\frac{y^2}{x^2+xz+z^2}+\frac{z^2}{y^2+yx+x^2}\ge \frac{5}{3}(x^2+y^2+z^2)+2xyz-6.$

Published on 25 Mar 2026 - 18:50

score 41 · Answer 12 · 2026-03-26 04:34:31

41

Views

Find max $T={\sum\frac {1-2yz}{3\,{y}^{2}+3\,{z}^{2}+{x}^{2}}}.$

Published on 26 Mar 2026 - 4:34

score 243 · Answer 13 · 2026-03-25 11:14:49

243

Views

Show any root of an elementary symmetric polynomial does not have positive imaginary part in all it's components

Published on 25 Mar 2026 - 11:14

score 131 · Answer 14 · 2026-03-26 04:35:12

131

Views

How to prove $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\ge \frac{3}{2}$ with pqr?

Published on 26 Mar 2026 - 4:35

score 54 · Answer 15 · 2026-03-25 05:59:11

54

Views

Simplifying polynomials in barycentric coordinates

Published on 25 Mar 2026 - 5:59

List Question

$(a^2+b^2+c^2+k(ab+bc+ca))(\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2})\geq \frac{9}{4}(2-k)$

How to prove $\frac{\sqrt{ab+bc+ca}}{2}\ge\frac{a^3(b+c)+b^3(c+a)+c^3(a+b)}{(a^2+b^2+c^2)(a+b+c)-2abc}.$?

If $x^2+y^2+z^2=1,$ prove $\frac{x(z^2-y^2+2y)}{z}+\frac{y(x^2-z^2+2z)}{x}+\frac{z(y^2-x^2+2x)}{y}\ge 2$

Finding $\small{\max\limits_{ab+bc+ca=3}\frac{ab}{2a+b}+\frac{bc}{2b+c}+\frac{ca}{2c+a}.}$

Find minimum $\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}$ when $a+b+c=4$

Finding $\small{\min\limits_{ab+bc+ca=3}\dfrac{1}{4a+bc}+\dfrac{1}{4b+ca}+\dfrac{1}{4c+ab}.}$

A difficult minimum

Let $a+b+c=3,$ then prove$\sqrt[3]{\frac{a+b}{5ab+4}}+\sqrt[3]{\frac{c+b}{5cb+4}}+\sqrt[3]{\frac{a+c}{5ac+4}}\ge \sqrt[3]{6}$

Finding $\small{\max\limits_{ab+bc+ca=abc+2}\frac{ab(2-c)}{a^2+abc+b^2}+\frac{bc(2-a)}{b^2+abc+c^2}+\frac{ca(2-b)}{c^2+abc+a^2}.}$

For $x+y+z=3,$ prove $\frac{1}{x^2+4}+\frac{1}{y^2+4}+\frac{1}{z^2+4}\le \frac{3}{5}.$

If $x,y,z>0: x+y+z=3,$ prove $\frac{x^2}{y^2+yz+z^2}+\frac{y^2}{x^2+xz+z^2}+\frac{z^2}{y^2+yx+x^2}\ge \frac{5}{3}(x^2+y^2+z^2)+2xyz-6.$

Find max $T={\sum\frac {1-2yz}{3\,{y}^{2}+3\,{z}^{2}+{x}^{2}}}.$

Show any root of an elementary symmetric polynomial does not have positive imaginary part in all it's components

How to prove $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\ge \frac{3}{2}$ with pqr?

Simplifying polynomials in barycentric coordinates

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