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 950 · Answer 2 · 2026-03-25 13:55:21

950

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Why are AM-GM inequality related exercises so popular

Published on 25 Mar 2026 - 13:55

score 106 · Answer 3 · 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 4 · 2026-03-26 04:52:27

1.3k

Views

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

Published on 26 Mar 2026 - 4:52

score 222 · Answer 5 · 2018-09-25 15:47:27

222

Views

Proving $\sum_{k=1}^n\vert\cos(k)\vert\ge \frac{n}{4}$ for all $n>0$

Published on 25 Sep 2018 - 15:47

score 160 · Answer 6 · 2026-03-26 04:52:22

160

Views

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

Published on 26 Mar 2026 - 4:52

score 167 · Answer 7 · 2018-09-25 17:33:10

167

Views

Finding all solutions to $cx \leq |x|$

Published on 25 Sep 2018 - 17:33

score 345 · Answer 8 · 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 94 · Answer 9 · 2026-03-28 17:41:20

94

Views

Proving $n \ge 41, \dfrac{2^{\lfloor\sqrt{2n}-3\rfloor}}{n} \ge 1$ without using derivatives

Published on 28 Mar 2026 - 17:41

score 3.6k · Answer 10 · 2026-03-30 10:42:30

3.6k

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Prove that the set $A := \left\{ (x,y) \in \Bbb R_{> 0}^2 \mid xy \geq 1 \right\}$ is convex

Published on 30 Mar 2026 - 10:42

score 997 · Answer 11 · 2026-04-02 01:56:25

997

Views

Prove for all positive real numbers $x$ and $y$, if $x+y \le (4xy)/(x+y)$, then $x=y$

Published on 02 Apr 2026 - 1:56

score 52 · Answer 12 · 2026-03-26 03:00:47

52

Views

Help with parametric equations and their inequalities

Published on 26 Mar 2026 - 3:00

score 153 · Answer 13 · 2018-09-26 05:25:17

153

Views

Bound the Frobenius norm of a matrix by largest entry and rank?

Published on 26 Sep 2018 - 5:25

score 273 · Answer 14 · 2026-03-28 04:25:23

273

Views

Inequality - Logarithm of the Sum of Exponentials

Published on 28 Mar 2026 - 4:25

score 112 · Answer 15 · 2026-03-25 12:48:45

112

Views

Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$?

Published on 25 Mar 2026 - 12:48

List Question

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

Why are AM-GM inequality related exercises so popular

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 $\log_5 6$?

Proving $\sum_{k=1}^n\vert\cos(k)\vert\ge \frac{n}{4}$ for all $n>0$

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

Finding all solutions to $cx \leq |x|$

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

Proving $n \ge 41, \dfrac{2^{\lfloor\sqrt{2n}-3\rfloor}}{n} \ge 1$ without using derivatives

Prove that the set $A := \left\{ (x,y) \in \Bbb R_{> 0}^2 \mid xy \geq 1 \right\}$ is convex

Prove for all positive real numbers $x$ and $y$, if $x+y \le (4xy)/(x+y)$, then $x=y$

Help with parametric equations and their inequalities

Bound the Frobenius norm of a matrix by largest entry and rank?

Inequality - Logarithm of the Sum of Exponentials

Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$?

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