Question

Minimize $(x+y)(y+z)(z+x)$ given $xyz(x+y+z) = 1$

score 351 · Answer 1 · 2026-03-26 17:34:31

351

Views

Minimize $(x+y)(y+z)(z+x)$ given $xyz(x+y+z) = 1$

Published on 26 Mar 2026 - 17:34

score 97 · Answer 2 · 2020-09-18 07:48:18

97

Views

If $a^2+b^2+c^2+d^2=4$ then $(a+2)(b+2)\geq cd$

Published on 18 Sep 2020 - 7:48

score 245 · Answer 3 · 2020-09-18 16:11:12

245

Views

Proving a polynomial is positive

Published on 18 Sep 2020 - 16:11

score 144 · Answer 4 · 2020-09-20 09:33:38

144

Views

Inequality $\frac{x_1}{x_k+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_k}{x_{k-1}+x_1}\geq 2.$

Published on 20 Sep 2020 - 9:33

score 125 · Answer 5 · 2020-09-21 09:22:48

125

Views

Minimum of a function without calculus. $a=\frac{{(1+t^2)}^3}{t^4}$

Published on 21 Sep 2020 - 9:22

score 78 · Answer 6 · 2026-03-27 10:10:52

78

Views

Prove the inequality without using the concept of Arithmetic and Geometric mean inequality

Published on 27 Mar 2026 - 10:10

score 2.1k · Answer 7 · 2026-03-25 23:22:52

2.1k

Views

Homogenization, what it is in inequalities and how to utilize it to its fullest.

Published on 25 Mar 2026 - 23:22

score 71 · Answer 8 · 2026-03-26 23:10:52

71

Views

question from Euclid 2011 about proving that $\frac{(a^3+b^3)c}{a^2+b^2}+\frac{(b^3+c^3)a}{b^2+c^2}+\frac{(c^3+a^3)}{c^2+a^2}\ge 1$

Published on 26 Mar 2026 - 23:10

score 68 · Answer 9 · 2020-09-23 07:47:31

68

Views

Minimum value of $f(x,y,z) = x^z + y^z - (xy)^{\frac{z}{4}}, x > 0, y > 0, z > 0$

Published on 23 Sep 2020 - 7:47

score 202 · Answer 10 · 2020-09-24 04:06:32

202

Views

Alternative approach for proving that for any $x\in\mathbb R^+$, $x^2+3x+\frac{1}{x} \ge \frac{15}{4}$.

Published on 24 Sep 2020 - 4:06

score 162 · Answer 11 · 2020-09-24 16:03:32

162

Views

If $a^2x^4+b^2y^4=c^6$, then the maximum value of $xy$

Published on 24 Sep 2020 - 16:03

score 120 · Answer 12 · 2020-09-24 23:03:36

120

Views

$\frac{a}{a^{2}+b^{2}+2}+\frac{b}{b^{2}+c^{2}+2}+\frac{c}{c^{2}+d^{2}+2}+\frac{d}{d^{2}+a^{2}+2}\le 1$

Published on 24 Sep 2020 - 23:03

score 151 · Answer 13 · 2026-03-27 01:00:04

151

Views

Show that $\int_0^1 f^3(x) dx + \frac{4}{27} \ge \left( \int_0^1 f(x) dx \right)^2$, where $f',f'' >0$

Published on 27 Mar 2026 - 1:00

score 155 · Answer 14 · 2026-03-26 23:10:58

155

Views

$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}$ for $a,b,c\in\mathbb{R}^+$ with $abc=1$

Published on 26 Mar 2026 - 23:10

score 42 · Answer 15 · 2020-09-29 17:34:53

42

Views

$\sqrt{k}+\frac{N}{k}\geq N^{1/3}$ where $N, k\geq 1$.

Published on 29 Sep 2020 - 17:34

List Question

Minimize $(x+y)(y+z)(z+x)$ given $xyz(x+y+z) = 1$

If $a^2+b^2+c^2+d^2=4$ then $(a+2)(b+2)\geq cd$

Proving a polynomial is positive

Inequality $\frac{x_1}{x_k+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_k}{x_{k-1}+x_1}\geq 2.$

Minimum of a function without calculus. $a=\frac{{(1+t^2)}^3}{t^4}$

Prove the inequality without using the concept of Arithmetic and Geometric mean inequality

Homogenization, what it is in inequalities and how to utilize it to its fullest.

question from Euclid 2011 about proving that $\frac{(a^3+b^3)c}{a^2+b^2}+\frac{(b^3+c^3)a}{b^2+c^2}+\frac{(c^3+a^3)}{c^2+a^2}\ge 1$

Minimum value of $f(x,y,z) = x^z + y^z - (xy)^{\frac{z}{4}}, x > 0, y > 0, z > 0$

Alternative approach for proving that for any $x\in\mathbb R^+$, $x^2+3x+\frac{1}{x} \ge \frac{15}{4}$.

If $a^2x^4+b^2y^4=c^6$, then the maximum value of $xy$

$\frac{a}{a^{2}+b^{2}+2}+\frac{b}{b^{2}+c^{2}+2}+\frac{c}{c^{2}+d^{2}+2}+\frac{d}{d^{2}+a^{2}+2}\le 1$

Show that $\int_0^1 f^3(x) dx + \frac{4}{27} \ge \left( \int_0^1 f(x) dx \right)^2$, where $f',f'' >0$

$\frac{a^3}{b+c}+\frac{b^3}{a+c}+\frac{c^3}{a+b}\geq \frac{3}{2}$ for $a,b,c\in\mathbb{R}^+$ with $abc=1$

$\sqrt{k}+\frac{N}{k}\geq N^{1/3}$ where $N, k\geq 1$.

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