Question

Intermediate inequality for an inequality from Crux mathematicorum

score 56 · Answer 1 · 2026-03-25 09:48:36

56

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Intermediate inequality for an inequality from Crux mathematicorum

Published on 25 Mar 2026 - 9:48

score 800 · Answer 2 · 2020-10-04 03:47:47

800

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

Published on 04 Oct 2020 - 3:47

score 144 · Answer 3 · 2020-10-21 12:46:09

144

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Prove $5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$

Published on 21 Oct 2020 - 12:46

score 99 · Answer 4 · 2020-10-23 10:04:22

99

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Prove $\sum_{\mathrm{cyc}} \sqrt{34x^2 + 28y^2 + 7z^2 - xy - 28yz + 41zx} \ge 9x + 9y + 9z$

Published on 23 Oct 2020 - 10:04

score 344 · Answer 5 · 2020-11-08 02:35:25

344

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Prove $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$ for $c = \min(a, b, c)$ and $ab + bc + ca = 2$

Published on 08 Nov 2020 - 2:35

score 92 · Answer 6 · 2020-11-09 08:28:10

92

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Prove $\sum\limits_{cyc} \frac{(a+14)^2}{(b+4)^2}\geq \frac{3}{2} \sum\limits_{cyc} \left(\frac{a+14}{b+4}+\frac{b+4}{a+14}\right)-6$

Published on 09 Nov 2020 - 8:28

score 499 · Answer 7 · 2020-12-21 08:56:44

499

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Minimizing number of multiplication signs in Computational linear algebra for kind of such these examples

Published on 21 Dec 2020 - 8:56

score 290 · Answer 8 · 2020-12-26 04:33:55

290

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Cyclic inequality with $a+b+c=1$

Published on 26 Dec 2020 - 4:33

score 27 · Answer 9 · 2021-01-10 11:38:45

27

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Disclude a number in a sequence so that the greatest divisor common of all the others at most. Find that GCD and the position of that discluded one.

Published on 10 Jan 2021 - 11:38

score 405 · Answer 10 · 2021-02-22 12:53:18

405

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A travelled inequality found by discriminant

Published on 22 Feb 2021 - 12:53

score 493 · Answer 11 · 2021-02-26 13:37:25

493

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$\sqrt{\frac{a^2+4bc}{b^2+c^2}}+\sqrt{\frac{b^2+4ac}{a^2+c^2}}+\sqrt{\frac{c^2+4ba}{b^2+a^2}}\ge 2+\sqrt{2}$

Published on 26 Feb 2021 - 13:37

score 97 · Answer 12 · 2021-02-28 15:33:56

97

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How to show that the constraint $(C)$ exists?

Published on 28 Feb 2021 - 15:33

score 77 · Answer 13 · 2021-09-11 21:58:18

77

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For any positive real numbers $a,b,c$, show that $\min\{(b-c)^2,(c-a)^2,(a-b)^2)\} \leq \frac{a^2+b^2+c^2}{5}$

Published on 11 Sep 2021 - 21:58

score 166 · Answer 14 · 2021-10-16 14:59:28

166

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How to prove $\frac{ab^2}{1+2b^2+c^2}+\frac{bc^2}{1+2c^2+a^2}+\frac{ca^2}{1+2a^2+b^2} \le \frac{3}{4}$ if $a+b+c=3$

Published on 16 Oct 2021 - 14:59

score 90 · Answer 15 · 2021-12-05 14:54:58

90

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Prove inequality using a given inequality

Published on 05 Dec 2021 - 14:54

List Question

Intermediate inequality for an inequality from Crux mathematicorum

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

Prove $5\Big(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\Big)\geq \frac{a^2+b^2+c^2}{ab+bc+ca}+10.$

Prove $\sum_{\mathrm{cyc}} \sqrt{34x^2 + 28y^2 + 7z^2 - xy - 28yz + 41zx} \ge 9x + 9y + 9z$

Prove $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$ for $c = \min(a, b, c)$ and $ab + bc + ca = 2$

Prove $\sum\limits_{cyc} \frac{(a+14)^2}{(b+4)^2}\geq \frac{3}{2} \sum\limits_{cyc} \left(\frac{a+14}{b+4}+\frac{b+4}{a+14}\right)-6$

Minimizing number of multiplication signs in Computational linear algebra for kind of such these examples

Cyclic inequality with $a+b+c=1$

Disclude a number in a sequence so that the greatest divisor common of all the others at most. Find that GCD and the position of that discluded one.

A travelled inequality found by discriminant

$\sqrt{\frac{a^2+4bc}{b^2+c^2}}+\sqrt{\frac{b^2+4ac}{a^2+c^2}}+\sqrt{\frac{c^2+4ba}{b^2+a^2}}\ge 2+\sqrt{2}$

How to show that the constraint $(C)$ exists?

For any positive real numbers $a,b,c$, show that $\min\{(b-c)^2,(c-a)^2,(a-b)^2)\} \leq \frac{a^2+b^2+c^2}{5}$

How to prove $\frac{ab^2}{1+2b^2+c^2}+\frac{bc^2}{1+2c^2+a^2}+\frac{ca^2}{1+2a^2+b^2} \le \frac{3}{4}$ if $a+b+c=3$

Prove inequality using a given inequality

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