Question

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

score 84 · Answer 1 · 2026-03-26 19:20:07

84

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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 26 Mar 2026 - 19:20

score 81 · Answer 2 · 2020-10-27 13:41:37

81

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Prove inequality $|a-b+c-d| \leqslant \frac{1}{16}$

Published on 27 Oct 2020 - 13:41

score 251 · Answer 3 · 2020-10-27 19:04:43

251

Views

Solving the system $\tan x+\sin y+\sin z=3x$, $\sin x+\tan y+\sin z=3y$, $\sin x+\sin y+\tan z=3z$

Published on 27 Oct 2020 - 19:04

score 118 · Answer 4 · 2026-03-25 23:24:47

118

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$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$ Proof

Published on 25 Mar 2026 - 23:24

score 101 · Answer 5 · 2020-10-29 20:57:16

101

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$(a+b)^2\le(1+c)a^2+(1+\frac{1}{c})b^2$ Proof

Published on 29 Oct 2020 - 20:57

score 111 · Answer 6 · 2020-10-31 19:30:09

111

Views

Minimize $x^2y^2z^2$ using classical inequalities.

Published on 31 Oct 2020 - 19:30

score 226 · Answer 7 · 2020-11-01 11:10:26

226

Views

Deriving AM-GM in a way that reflects my intuition.

Published on 01 Nov 2020 - 11:10

score 341 · Answer 8 · 2020-11-02 01:02:52

341

Views

Prove that $\frac{(1-x_1)(1-x_2) \cdots (1-x_n)}{x_1 x_2 \cdots x_n} \geq (n-1)^n$ for $x_i > 0$ and $x_1 + \ldots + x_n = 1$

Published on 02 Nov 2020 - 1:02

score 75 · Answer 9 · 2020-11-05 16:06:35

75

Views

Prove that $x ^ {1 - x} (1 - x) ^ {x} \le \frac {1}{2}$

Published on 05 Nov 2020 - 16:06

score 276 · Answer 10 · 2020-11-07 13:07:06

276

Views

Partial Proof of an inequality by Vasile Cirtoaje

Published on 07 Nov 2020 - 13:07

score 75 · Answer 11 · 2020-11-12 14:17:20

75

Views

For $a,b>0$ and $c\ge0$ prove that $\frac{2(a+b)}{3a+6b+9c}+\frac{6(b+c)}{5a+2b+3c}+\frac{3(a+c)}{2a+8b+6c}\ge\frac{3}{2}$

Published on 12 Nov 2020 - 14:17

score 202 · Answer 12 · 2020-11-15 17:23:02

202

Views

Finding maximum possible value

Published on 15 Nov 2020 - 17:23

score 61 · Answer 13 · 2020-11-17 19:51:03

61

Views

For $a,b,c>0$, such that: $a^{\frac{3}{2}}+b^{\frac{3}{2}}+c^{\frac{3}{2}}=3$ prove that:$(a+b)^3+(b+c)^3+(a+c)^3\le24$

Published on 17 Nov 2020 - 19:51

score 102 · Answer 14 · 2026-03-26 23:10:35

102

Views

Inequality $\frac{xy+z}{x+yz}+\frac{yz+x}{y+zx}+\frac{zx+y}{z+xy}-\frac{x+y+z}{3}\leq 1$

Published on 26 Mar 2026 - 23:10

score 151 · Answer 15 · 2020-11-18 12:45:03

151

Views

Inequality with mean inequality

Published on 18 Nov 2020 - 12:45

List Question

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

Prove inequality $|a-b+c-d| \leqslant \frac{1}{16}$

Solving the system $\tan x+\sin y+\sin z=3x$, $\sin x+\tan y+\sin z=3y$, $\sin x+\sin y+\tan z=3z$

$\frac{b^2-a^2}{c+a} + \frac{c^2-b^2}{a+b} + \frac{a^2-c^2}{b+c} \ge 0$ Proof

$(a+b)^2\le(1+c)a^2+(1+\frac{1}{c})b^2$ Proof

Minimize $x^2y^2z^2$ using classical inequalities.

Deriving AM-GM in a way that reflects my intuition.

Prove that $\frac{(1-x_1)(1-x_2) \cdots (1-x_n)}{x_1 x_2 \cdots x_n} \geq (n-1)^n$ for $x_i > 0$ and $x_1 + \ldots + x_n = 1$

Prove that $x ^ {1 - x} (1 - x) ^ {x} \le \frac {1}{2}$

Partial Proof of an inequality by Vasile Cirtoaje

For $a,b>0$ and $c\ge0$ prove that $\frac{2(a+b)}{3a+6b+9c}+\frac{6(b+c)}{5a+2b+3c}+\frac{3(a+c)}{2a+8b+6c}\ge\frac{3}{2}$

Finding maximum possible value

For $a,b,c>0$, such that: $a^{\frac{3}{2}}+b^{\frac{3}{2}}+c^{\frac{3}{2}}=3$ prove that:$(a+b)^3+(b+c)^3+(a+c)^3\le24$

Inequality $\frac{xy+z}{x+yz}+\frac{yz+x}{y+zx}+\frac{zx+y}{z+xy}-\frac{x+y+z}{3}\leq 1$

Inequality with mean inequality

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