Question

$\sqrt{a^2+5b^2}+\sqrt{b^2+5c^2}+\sqrt{c^2+5a^2}\geq\sqrt{10(a^2+b^2+c^2)+8(ab+ac+bc)}$ for any real numbers.

score 801 · Answer 1 · 2026-03-26 01:23:33

801

Views

$\sqrt{a^2+5b^2}+\sqrt{b^2+5c^2}+\sqrt{c^2+5a^2}\geq\sqrt{10(a^2+b^2+c^2)+8(ab+ac+bc)}$ for any real numbers.

Published on 26 Mar 2026 - 1:23

score 200 · Answer 2 · 2020-10-07 06:36:41

200

Views

prove that $\sum_{cyc}\frac{a}{b^2+c^2}\ge \frac{4}{5}\sum_{cyc}\frac{1}{b+c}$

Published on 07 Oct 2020 - 6:36

score 116 · Answer 3 · 2020-10-23 14:30:46

116

Views

Prove $3\left(9-5\sqrt{3}\right) \sum \frac{1}{a} \geqslant \sum a^2+\frac32\cdot\frac{\left[(\sqrt3-2)(ab+bc+ca)+abc\right]^2}{abc}$

Published on 23 Oct 2020 - 14:30

score 83 · Answer 4 · 2020-10-25 14:30:11

83

Views

Find maximize of $P=\frac{x\sqrt{yz}}{\sqrt{x^2+1}\sqrt[4]{\left(y^2+4\right)\left(z^2+9\right)}}$

Published on 25 Oct 2020 - 14:30

score 210 · Answer 5 · 2021-01-02 13:59:00

210

Views

Hard refinement concerning the inequality $\frac{1}{7a+b}+\frac{1}{7b+c}+\frac{1}{7c+a}\leq\frac{\sqrt{3}}{8}$

Published on 02 Jan 2021 - 13:59

score 486 · Answer 6 · 2021-01-02 21:53:05

486

Views

On the Abstract Concreteness Method (bka $ABC-$Method).

Published on 02 Jan 2021 - 21:53

score 174 · Answer 7 · 2021-02-17 03:55:39

174

Views

Prove that $\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\ge a+b+c$

Published on 17 Feb 2021 - 3:55

score 75 · Answer 8 · 2021-03-12 07:02:59

75

Views

About a cyclic expression problem

Published on 12 Mar 2021 - 7:02

score 601 · Answer 9 · 2021-03-19 22:11:52

601

Views

The inequality $\,2+\sqrt{\frac p2}\leq\sum\limits_\text{cyc}\sqrt{\frac{a^2+pbc}{b^2+c^2}}\,$ where $0\leq p\leq 2$ is: Probably true! Provably true?

Published on 19 Mar 2021 - 22:11

score 170 · Answer 10 · 2021-04-02 12:07:36

170

Views

Conjecture-refinement related to Nesbitt's inequality

Published on 02 Apr 2021 - 12:07

score 121 · Answer 11 · 2026-03-25 15:41:23

121

Views

Prove that: $(a^5-2a+4)(b^5-2b+4)(c^5-2c+4)\ge9(ab+bc+ca)$

Published on 25 Mar 2026 - 15:41

score 63 · Answer 12 · 2021-07-02 03:43:14

63

Views

With the $uwv$ method, why can $3v^2$ be negative?

Published on 02 Jul 2021 - 3:43

score 162 · Answer 13 · 2021-07-02 19:12:49

162

Views

Prove that: $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+5\ge (a+b)(b+c)(c+a)$

Published on 02 Jul 2021 - 19:12

score 187 · Answer 14 · 2026-03-26 12:08:30

187

Views

An inequality with $x,y,z \ge -1$ and $x+y+z=1$

Published on 26 Mar 2026 - 12:08

score 105 · Answer 15 · 2021-08-01 17:46:58

105

Views

Hard and non-trivial inequality with three variable reals

Published on 01 Aug 2021 - 17:46

List Question

$\sqrt{a^2+5b^2}+\sqrt{b^2+5c^2}+\sqrt{c^2+5a^2}\geq\sqrt{10(a^2+b^2+c^2)+8(ab+ac+bc)}$ for any real numbers.

prove that $\sum_{cyc}\frac{a}{b^2+c^2}\ge \frac{4}{5}\sum_{cyc}\frac{1}{b+c}$

Prove $3\left(9-5\sqrt{3}\right) \sum \frac{1}{a} \geqslant \sum a^2+\frac32\cdot\frac{\left[(\sqrt3-2)(ab+bc+ca)+abc\right]^2}{abc}$

Find maximize of $P=\frac{x\sqrt{yz}}{\sqrt{x^2+1}\sqrt[4]{\left(y^2+4\right)\left(z^2+9\right)}}$

Hard refinement concerning the inequality $\frac{1}{7a+b}+\frac{1}{7b+c}+\frac{1}{7c+a}\leq\frac{\sqrt{3}}{8}$

On the Abstract Concreteness Method (bka $ABC-$Method).

Prove that $\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\ge a+b+c$

About a cyclic expression problem

The inequality $\,2+\sqrt{\frac p2}\leq\sum\limits_\text{cyc}\sqrt{\frac{a^2+pbc}{b^2+c^2}}\,$ where $0\leq p\leq 2$ is: Probably true! Provably true?

Conjecture-refinement related to Nesbitt's inequality

Prove that: $(a^5-2a+4)(b^5-2b+4)(c^5-2c+4)\ge9(ab+bc+ca)$

With the $uwv$ method, why can $3v^2$ be negative?

Prove that: $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+5\ge (a+b)(b+c)(c+a)$

An inequality with $x,y,z \ge -1$ and $x+y+z=1$

Hard and non-trivial inequality with three variable reals

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