Question

If $abc=1$ where $a,b,c>0$, then show that $(a-1+b^{-1})(b-1+c^{-1})(c-1+a^{-1}) \leq 1$.

score 133 · Answer 1 · 2020-08-08 03:20:16

133

Views

If $abc=1$ where $a,b,c>0$, then show that $(a-1+b^{-1})(b-1+c^{-1})(c-1+a^{-1}) \leq 1$.

Published on 08 Aug 2020 - 3:20

score 101 · Answer 2 · 2026-03-25 19:10:19

101

Views

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

Published on 25 Mar 2026 - 19:10

score 114 · Answer 3 · 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 4 · 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 69 · Answer 5 · 2020-08-15 07:07:32

69

Views

What topic in algebraic manipulation does this fall under?

Published on 15 Aug 2020 - 7:07

score 261 · Answer 6 · 2020-08-17 23:05:10

261

Views

Decompose the symmetric polynomial $\Sigma{x_1^2x_2^2x_3^2}$ into elementary symmetric polynomials.

Published on 17 Aug 2020 - 23:05

score 123 · Answer 7 · 2026-03-26 17:53:53

123

Views

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

Published on 26 Mar 2026 - 17:53

score 116 · Answer 8 · 2026-03-26 14:22:58

116

Views

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 26 Mar 2026 - 14:22

score 77 · Answer 9 · 2020-08-20 06:24:03

77

Views

Does $(x-1)^2+(y-1)^2 \le c\big((x-y)^2+(xy-1)^2\big) $ hold?

Published on 20 Aug 2020 - 6:24

score 52 · Answer 10 · 2026-03-26 21:27:07

52

Views

A (additive) basis for the group of homogeneous symmetric polynomials (with integer coefficients) of degree $k$ in $n$ variables

Published on 26 Mar 2026 - 21:27

score 338 · Answer 11 · 2026-03-25 19:10:48

338

Views

Two inequalities with parameters $a,b,c>0$ such that $ca+ab+bc+abc\leq 4$

Published on 25 Mar 2026 - 19:10

score 133 · Answer 12 · 2026-03-25 19:10:26

133

Views

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

Published on 25 Mar 2026 - 19:10

score 237 · Answer 13 · 2026-03-26 22:53:48

237

Views

Proving $3(1−a+a^2)(1−b+b^2)(1−c+c^2)≥1+abc+a^2b^2c^2$

Published on 26 Mar 2026 - 22:53

score 807 · Answer 14 · 2026-03-26 20:44:32

807

Views

Can every symmetric function be written as some function of a sum?

Published on 26 Mar 2026 - 20:44

score 140 · Answer 15 · 2026-03-26 21:26:07

140

Views

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

Published on 26 Mar 2026 - 21:26

List Question

If $abc=1$ where $a,b,c>0$, then show that $(a-1+b^{-1})(b-1+c^{-1})(c-1+a^{-1}) \leq 1$.

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} $

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

What topic in algebraic manipulation does this fall under?

Decompose the symmetric polynomial $\Sigma{x_1^2x_2^2x_3^2}$ into elementary symmetric polynomials.

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

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

Does $(x-1)^2+(y-1)^2 \le c\big((x-y)^2+(xy-1)^2\big) $ hold?

A (additive) basis for the group of homogeneous symmetric polynomials (with integer coefficients) of degree $k$ in $n$ variables

Two inequalities with parameters $a,b,c>0$ such that $ca+ab+bc+abc\leq 4$

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

Proving $3(1−a+a^2)(1−b+b^2)(1−c+c^2)≥1+abc+a^2b^2c^2$

Can every symmetric function be written as some function of a sum?

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

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