Question

Prove $1<ab+bc+ca-abc<\frac{28}{27}$

score 131 · Answer 1 · 2022-01-03 09:46:03

131

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Prove $1<ab+bc+ca-abc<\frac{28}{27}$

Published on 03 Jan 2022 - 9:46

score 132 · Answer 2 · 2022-01-05 16:51:51

132

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(Dis)prove the inequality $2^{\frac{1}{2}}\cdot\left(f\left(xa\right)+f\left(\frac{1}{xa}\right)\right)^{\frac{1}{2}}\leq g(x)$ for $a,x>0$

Published on 05 Jan 2022 - 16:51

score 216 · Answer 3 · 2022-01-08 11:47:53

216

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Prove that $\displaystyle{\prod_{k=1}^n \frac{1+x_k}{x_k} \geq \prod_{k=1}^n \frac{n-x_k}{1-x_k}}$

Published on 08 Jan 2022 - 11:47

score 54 · Answer 4 · 2022-01-13 21:43:14

54

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Prove $\frac{4b^2+b+1}{4\left|b\right|}\ge \sqrt{b+1}$, $b\ge -1,\:b\ne 0$

Published on 13 Jan 2022 - 21:43

score 130 · Answer 5 · 2022-01-14 01:56:06

130

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Let $a,b\in \mathbb{R}$ such that $ab=2$. Find the max value of $\frac{3}{2\left(a+b\right)^2}$ and $a, b$ where max is attained, without calculus.

Published on 14 Jan 2022 - 1:56

score 91 · Answer 6 · 2022-01-14 22:17:52

91

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Find $a$ such that ${x_1}^2+{x_2}^2$ takes the minimal value where $x_1, x_2$ are solutions to $x^2-ax+(a-1)=0$ DO NOT USE CALCULUS

Published on 14 Jan 2022 - 22:17

score 97 · Answer 7 · 2022-01-15 05:27:44

97

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How to prove $\dfrac{a+b}{2}\geq\sqrt{ab}$ using ellipse

Published on 15 Jan 2022 - 5:27

score 120 · Answer 8 · 2022-01-25 11:46:05

120

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$a + b + c + d = 1$ and $a^2 + b^2 + c^2 + d^2 = \frac{1}{3}$, where $-1 \le a,b,c,d \le 1$. Which value of $a$ is the largest possible?

Published on 25 Jan 2022 - 11:46

score 48 · Answer 9 · 2022-02-06 09:06:52

48

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What is the fastest elementary proof of $A_n≥G_n$?

Published on 06 Feb 2022 - 9:06

score 394 · Answer 10 · 2022-02-06 14:15:36

394

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Find the maximum value of $x^2y^3$ subject to the condition that $3x+2y=1$

Published on 06 Feb 2022 - 14:15

score 98 · Answer 11 · 2022-02-08 23:38:58

98

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A friends proof of AM-GM inequality

Published on 08 Feb 2022 - 23:38

score 95 · Answer 12 · 2022-02-09 16:06:03

95

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Show that $x^x(1-x)^{(1-x)} \geq \frac{1}{2}$ for $x \in (0,1)$.

Published on 09 Feb 2022 - 16:06

score 71 · Answer 13 · 2022-02-11 12:21:51

71

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Proof that $\frac{x}{\sqrt y}+\frac{y}{\sqrt z}+\frac{z}{\sqrt x} \geq 3$ for $x+y+z=3$

Published on 11 Feb 2022 - 12:21

score 47 · Answer 14 · 2022-02-17 04:34:31

47

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question about AM-GM inequality ab<(a^2+b^2)/2

Published on 17 Feb 2022 - 4:34

score 116 · Answer 15 · 2022-02-24 00:30:26

116

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Hard AM-GM Inequality

Published on 24 Feb 2022 - 0:30

List Question

Prove $1<ab+bc+ca-abc<\frac{28}{27}$

(Dis)prove the inequality $2^{\frac{1}{2}}\cdot\left(f\left(xa\right)+f\left(\frac{1}{xa}\right)\right)^{\frac{1}{2}}\leq g(x)$ for $a,x>0$

Prove that $\displaystyle{\prod_{k=1}^n \frac{1+x_k}{x_k} \geq \prod_{k=1}^n \frac{n-x_k}{1-x_k}}$

Prove $\frac{4b^2+b+1}{4\left|b\right|}\ge \sqrt{b+1}$, $b\ge -1,\:b\ne 0$

Let $a,b\in \mathbb{R}$ such that $ab=2$. Find the max value of $\frac{3}{2\left(a+b\right)^2}$ and $a, b$ where max is attained, without calculus.

Find $a$ such that ${x_1}^2+{x_2}^2$ takes the minimal value where $x_1, x_2$ are solutions to $x^2-ax+(a-1)=0$ DO NOT USE CALCULUS

How to prove $\dfrac{a+b}{2}\geq\sqrt{ab}$ using ellipse

$a + b + c + d = 1$ and $a^2 + b^2 + c^2 + d^2 = \frac{1}{3}$, where $-1 \le a,b,c,d \le 1$. Which value of $a$ is the largest possible?

What is the fastest elementary proof of $A_n≥G_n$?

Find the maximum value of $x^2y^3$ subject to the condition that $3x+2y=1$

A friends proof of AM-GM inequality

Show that $x^x(1-x)^{(1-x)} \geq \frac{1}{2}$ for $x \in (0,1)$.

Proof that $\frac{x}{\sqrt y}+\frac{y}{\sqrt z}+\frac{z}{\sqrt x} \geq 3$ for $x+y+z=3$

question about AM-GM inequality ab<(a^2+b^2)/2

Hard AM-GM Inequality

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