Question

How to prove $\frac{(x+y)(y+z)(z+x)}{4xyz}≥\frac{x+z}{y+z}+\frac{y+z}{x+z}$ for $x,y,z>0$?

score 79 · Answer 1 · 2019-02-01 15:57:36

79

Views

How to prove $\frac{(x+y)(y+z)(z+x)}{4xyz}≥\frac{x+z}{y+z}+\frac{y+z}{x+z}$ for $x,y,z>0$?

Published on 01 Feb 2019 - 15:57

score 48 · Answer 2 · 2019-02-04 07:48:37

48

Views

minimum value of $l$

Published on 04 Feb 2019 - 7:48

score 42 · Answer 3 · 2019-02-04 15:55:15

42

Views

Show for any k that $\|x\|^2\|y\|^2 -( \textbf{x} \cdot \textbf{y})^2 = \frac{1}{2} \sum_{i, j=1}^k (x_i y_j - x_j y_i)^2$

Published on 04 Feb 2019 - 15:55

score 977 · Answer 4 · 2026-03-25 09:22:07

977

Views

determining if sequence has upper bound

Published on 25 Mar 2026 - 9:22

score 121 · Answer 5 · 2026-03-26 17:53:32

121

Views

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Published on 26 Mar 2026 - 17:53

score 62 · Answer 6 · 2026-03-25 01:17:12

62

Views

Inequality using lengths of the edges of a triangle

Published on 25 Mar 2026 - 1:17

score 126 · Answer 7 · 2026-03-25 06:12:00

126

Views

Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$.

Published on 25 Mar 2026 - 6:12

score 163 · Answer 8 · 2026-03-26 05:56:20

163

Views

Inequality of cyclic expression

Published on 26 Mar 2026 - 5:56

score 210 · Answer 9 · 2019-02-09 01:51:42

210

Views

Proving $\sum\sqrt{A_i^2+B_i^2} \geq \sqrt{\left(\sum A_i\right)^2+\left(\sum B_i\right)^2} $

Published on 09 Feb 2019 - 1:51

score 128 · Answer 10 · 2026-03-25 06:13:06

128

Views

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

Published on 25 Mar 2026 - 6:13

score 49 · Answer 11 · 2026-03-25 09:23:31

49

Views

Using a Euclidean norm to bound a $k$-tuple

Published on 25 Mar 2026 - 9:23

score 127 · Answer 12 · 2026-03-26 06:01:37

127

Views

If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$?

Published on 26 Mar 2026 - 6:01

score 117 · Answer 13 · 2026-03-26 07:33:16

117

Views

Showing an inequality using Cauchy-Schwarz

Published on 26 Mar 2026 - 7:33

score 269 · Answer 14 · 2026-03-26 17:53:33

269

Views

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that?

Published on 26 Mar 2026 - 17:53

score 573 · Answer 15 · 2019-02-19 23:03:28

573

Views

Exercise 5.12 from Casella’s Book

Published on 19 Feb 2019 - 23:03

List Question

How to prove $\frac{(x+y)(y+z)(z+x)}{4xyz}≥\frac{x+z}{y+z}+\frac{y+z}{x+z}$ for $x,y,z>0$?

minimum value of $l$

Show for any k that $\|x\|^2\|y\|^2 -( \textbf{x} \cdot \textbf{y})^2 = \frac{1}{2} \sum_{i, j=1}^k (x_i y_j - x_j y_i)^2$

determining if sequence has upper bound

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Inequality using lengths of the edges of a triangle

Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$.

Inequality of cyclic expression

Proving $\sum\sqrt{A_i^2+B_i^2} \geq \sqrt{\left(\sum A_i\right)^2+\left(\sum B_i\right)^2} $

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

Using a Euclidean norm to bound a $k$-tuple

If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$?

Showing an inequality using Cauchy-Schwarz

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that?

Exercise 5.12 from Casella’s Book

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