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 82 · Answer 1 · 2026-04-11 20:11:12

82

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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 11 Apr 2026 - 20:11

score 51 · Answer 2 · 2026-04-14 08:26:41

51

Views

minimum value of $l$

Published on 14 Apr 2026 - 8:26

score 45 · Answer 3 · 2026-04-17 12:25:53

45

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 17 Apr 2026 - 12:25

score 980 · Answer 4 · 2026-05-11 02:37:29

980

Views

determining if sequence has upper bound

Published on 11 May 2026 - 2:37

score 123 · Answer 5 · 2026-04-17 00:07:07

123

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 17 Apr 2026 - 0:07

score 66 · Answer 6 · 2026-05-10 12:38:27

66

Views

Inequality using lengths of the edges of a triangle

Published on 10 May 2026 - 12:38

score 127 · Answer 7 · 2026-05-10 16:13:04

127

Views

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

Published on 10 May 2026 - 16:13

score 165 · Answer 8 · 2026-04-13 14:38:45

165

Views

Inequality of cyclic expression

Published on 13 Apr 2026 - 14:38

score 213 · Answer 9 · 2026-04-16 11:04:50

213

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 16 Apr 2026 - 11:04

score 129 · Answer 10 · 2026-05-10 13:51:38

129

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 10 May 2026 - 13:51

score 51 · Answer 11 · 2026-04-10 16:49:13

51

Views

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

Published on 10 Apr 2026 - 16:49

score 129 · Answer 12 · 2026-04-17 12:52:27

129

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 17 Apr 2026 - 12:52

score 119 · Answer 13 · 2026-04-12 21:51:42

119

Views

Showing an inequality using Cauchy-Schwarz

Published on 12 Apr 2026 - 21:51

score 271 · Answer 14 · 2026-04-16 15:13:59

271

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 16 Apr 2026 - 15:13

score 575 · Answer 15 · 2026-04-15 07:19:13

575

Views

Exercise 5.12 from Casella’s Book

Published on 15 Apr 2026 - 7:19

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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