Question

For positve $a$, $b$, $c$, $d$ with $a+b+c+d\leq 1$, prove that $\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq255\cdot a b c d .$

score 206 · Answer 1 · 2026-03-25 15:42:34

206

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For positve $a$, $b$, $c$, $d$ with $a+b+c+d\leq 1$, prove that $\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq255\cdot a b c d .$

Published on 25 Mar 2026 - 15:42

score 106 · Answer 2 · 2026-03-25 16:03:07

106

Views

How to prove inequality $(a^2+b^2+c^2)^3\ge6(a^3+b^3+c^3)^2$ when $a+b+c=0$?

Published on 25 Mar 2026 - 16:03

score 1.3k · Answer 3 · 2018-09-25 15:39:46

1.3k

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How to compare logarithms $\log_4 5$ and $\log_5 6$?

Published on 25 Sep 2018 - 15:39

score 159 · Answer 4 · 2018-09-25 16:08:31

159

Views

A very strange algebra problem with four numbers with an average 1.

Published on 25 Sep 2018 - 16:08

score 345 · Answer 5 · 2026-03-25 01:40:04

345

Views

For $a\geq2$, $b\geq2$ and $c\geq2$, prove that $\left(a^3+b\right)\left(b^3+c\right)\left(c^3+a\right)\geq125 abc$

Published on 25 Mar 2026 - 1:40

score 30 · Answer 6 · 2018-09-26 19:54:21

30

Views

Why do we usually require $b<a$ in A.M.-G.M. Inequality?

Published on 26 Sep 2018 - 19:54

score 68 · Answer 7 · 2018-10-03 13:03:22

68

Views

Two versions of AM-GM inequality

Published on 03 Oct 2018 - 13:03

score 93 · Answer 8 · 2026-03-25 04:40:28

93

Views

How to prove the given inequality using CS or AM GM HM inequality

Published on 25 Mar 2026 - 4:40

score 65 · Answer 9 · 2026-03-25 20:53:27

65

Views

$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$ for real $x,y$ with $x\in(0,1)$

Published on 25 Mar 2026 - 20:53

score 148 · Answer 10 · 2026-03-25 19:06:20

148

Views

Prove $\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$

Published on 25 Mar 2026 - 19:06

score 182 · Answer 11 · 2018-10-06 14:43:22

182

Views

Find the minimum values of $a^3$ and $b^3$ from the following given cubic function.

Published on 06 Oct 2018 - 14:43

score 75 · Answer 12 · 2026-03-25 01:35:34

75

Views

problem regarding application of Jensen's inequality

Published on 25 Mar 2026 - 1:35

score 405 · Answer 13 · 2026-03-25 18:47:16

405

Views

Maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ without using calculus

Published on 25 Mar 2026 - 18:47

score 96 · Answer 14 · 2018-10-15 00:54:06

96

Views

Making a proof using AM-GM.

Published on 15 Oct 2018 - 0:54

score 124 · Answer 15 · 2018-10-15 16:20:18

124

Views

Let $x$, $y$, $z$, $t$ are real numbers. $x+y+z+t=6$ and $\sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8$ Find each value of them as a number.

Published on 15 Oct 2018 - 16:20

List Question

For positve $a$, $b$, $c$, $d$ with $a+b+c+d\leq 1$, prove that $\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq255\cdot a b c d .$

How to prove inequality $(a^2+b^2+c^2)^3\ge6(a^3+b^3+c^3)^2$ when $a+b+c=0$?

How to compare logarithms $\log_4 5$ and $\log_5 6$?

A very strange algebra problem with four numbers with an average 1.

For $a\geq2$, $b\geq2$ and $c\geq2$, prove that $\left(a^3+b\right)\left(b^3+c\right)\left(c^3+a\right)\geq125 abc$

Why do we usually require $b<a$ in A.M.-G.M. Inequality?

Two versions of AM-GM inequality

How to prove the given inequality using CS or AM GM HM inequality

$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$ for real $x,y$ with $x\in(0,1)$

Prove $\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$

Find the minimum values of $a^3$ and $b^3$ from the following given cubic function.

problem regarding application of Jensen's inequality

Maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ without using calculus

Making a proof using AM-GM.

Let $x$, $y$, $z$, $t$ are real numbers. $x+y+z+t=6$ and $\sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8$ Find each value of them as a number.

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