Question

If $x,y,z\in\mathbb R^+$ are in Harmonic Progression, prove that $z\cdot e^{x-y}+x\cdot e^{z-y}\ge\frac{2xz}y$

score 43 · Answer 1 · 2026-03-28 22:29:56

43

Views

If $x,y,z\in\mathbb R^+$ are in Harmonic Progression, prove that $z\cdot e^{x-y}+x\cdot e^{z-y}\ge\frac{2xz}y$

Published on 28 Mar 2026 - 22:29

score 49 · Answer 2 · 2023-05-22 14:24:41

49

Views

Given the sum of $k$ numbers, what is the maximum value of their product?

Published on 22 May 2023 - 14:24

score 150 · Answer 3 · 2023-05-22 20:01:30

150

Views

Find the maximum of $\frac {xyz}{(1+x)(x+y)(y+z)(z+16)}$, given x, y, z are positive real numbers.

Published on 22 May 2023 - 20:01

score 69 · Answer 4 · 2023-05-23 19:01:02

69

Views

Prove or disprove the inequality for the length of the sides of the triangle $a, b, c$.

Published on 23 May 2023 - 19:01

score 25 · Answer 5 · 2023-05-26 17:13:25

25

Views

Prove $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} + \frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

Published on 26 May 2023 - 17:13

score 97 · Answer 6 · 2023-05-27 14:06:26

97

Views

Prove that for positive numbers x and y the inequality is true.

Published on 27 May 2023 - 14:06

score 101 · Answer 7 · 2023-05-27 15:52:00

101

Views

Prove $\sqrt{\frac{x}{\left(x+y\right)\left(1+y\right)}}+\sqrt{\frac{y}{\left(x+y\right)\left(1+x\right)}}+\sqrt{\frac{xy}{(1+x)(1+y)}}>1$

Published on 27 May 2023 - 15:52

score 100 · Answer 8 · 2023-05-28 11:46:22

100

Views

Prove $2(a^2+b^2+c^2)+\frac{4}{3}\sum\limits_{\mathrm{cyc}}\frac{1}{a^2+1} \geq 5$ for $ab+bc+ac=1$

Published on 28 May 2023 - 11:46

score 78 · Answer 9 · 2023-05-28 13:49:02

78

Views

Prove or disprove that the inequality is true if $xyz=1$ and $xyz=8^3$.

Published on 28 May 2023 - 13:49

score 41 · Answer 10 · 2023-06-12 14:20:31

41

Views

AM-GM inequality involving $a^n + b^n$

Published on 12 Jun 2023 - 14:20

score 120 · Answer 11 · 2023-06-20 05:10:26

120

Views

AM-GM inequality.

Published on 20 Jun 2023 - 5:10

score 68 · Answer 12 · 2023-06-21 02:25:08

Let $0 ≤ a, b, c, d ≤ 1$. Prove that at least one of the products $a(1 − b), b(1 − c), c(1 − d), d(1 − a)$ is $\leq \frac{1}{4}$ .

score 183 · Answer 13 · 2023-07-04 19:59:22

183

Views

What is the minimum value of the fraction $\frac{4x^4+4y^4-3xy+5}{2x^2+2y^2-xy}$

Published on 04 Jul 2023 - 19:59

score 121 · Answer 14 · 2026-03-26 04:35:14

121

Views

Show that $4(x^3 + y^3 + z^3 + 3) \geq 3(x+1)(y+1)(z+1)$ | Inequalities

Published on 26 Mar 2026 - 4:35

score 282 · Answer 15 · 2023-07-05 15:19:19

282

Views

Prove $\frac{1}{3} > \frac{2ab^2}{(a + \sqrt{a^2 + b^2})^3}$

Published on 05 Jul 2023 - 15:19

List Question

If $x,y,z\in\mathbb R^+$ are in Harmonic Progression, prove that $z\cdot e^{x-y}+x\cdot e^{z-y}\ge\frac{2xz}y$

Given the sum of $k$ numbers, what is the maximum value of their product?

Find the maximum of $\frac {xyz}{(1+x)(x+y)(y+z)(z+16)}$, given x, y, z are positive real numbers.

Prove or disprove the inequality for the length of the sides of the triangle $a, b, c$.

Prove $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} + \frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

Prove that for positive numbers x and y the inequality is true.

Prove $\sqrt{\frac{x}{\left(x+y\right)\left(1+y\right)}}+\sqrt{\frac{y}{\left(x+y\right)\left(1+x\right)}}+\sqrt{\frac{xy}{(1+x)(1+y)}}>1$

Prove $2(a^2+b^2+c^2)+\frac{4}{3}\sum\limits_{\mathrm{cyc}}\frac{1}{a^2+1} \geq 5$ for $ab+bc+ac=1$

Prove or disprove that the inequality is true if $xyz=1$ and $xyz=8^3$.

AM-GM inequality involving $a^n + b^n$

AM-GM inequality.

Let $0 ≤ a, b, c, d ≤ 1$. Prove that at least one of the products $a(1 − b), b(1 − c), c(1 − d), d(1 − a)$ is $\leq \frac{1}{4}$ .

What is the minimum value of the fraction $\frac{4x^4+4y^4-3xy+5}{2x^2+2y^2-xy}$

Show that $4(x^3 + y^3 + z^3 + 3) \geq 3(x+1)(y+1)(z+1)$ | Inequalities

Prove $\frac{1}{3} > \frac{2ab^2}{(a + \sqrt{a^2 + b^2})^3}$

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