Question

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

score 102 · Answer 1 · 2026-03-26 01:12:46

102

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Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

Published on 26 Mar 2026 - 1:12

score 93 · Answer 2 · 2026-03-26 19:20:09

93

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Find the minimize of $A=\frac{1}{a^2}\left(b^2+c^2\right)+a^2\left(\frac{1}{b^2}+\frac{1}{c^2}\right)$

Published on 26 Mar 2026 - 19:20

score 277 · Answer 3 · 2026-02-23 04:02:51

277

Views

Using the Mean Value Theorem, show that for all positive integers $n$, $n\ln(1+\frac{1}{n}) \leq 1$

Published on 23 Feb 2026 - 4:02

score 124 · Answer 4 · 2026-03-25 17:42:08

124

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Let $a,b,c$ be the length of the side $BC,CA,AB$ respectively for $\triangle ABC$. Show that $(2b+2c-2a)^3(a+b+c)\geq18a^2bc$.

Published on 25 Mar 2026 - 17:42

score 198 · Answer 5 · 2026-04-02 01:35:30

198

Views

Inequality from AMM problems section

Published on 02 Apr 2026 - 1:35

score 47 · Answer 6 · 2026-04-02 22:25:05

47

Views

Inequality in $n$-variables

Published on 02 Apr 2026 - 22:25

score 151 · Answer 7 · 2026-03-25 04:38:22

151

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What tools one should use for inequalities?

Published on 25 Mar 2026 - 4:38

score 4.4k · Answer 8 · 2026-03-25 00:02:21

4.4k

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When does the equality hold in the Holder inequality?

Published on 25 Mar 2026 - 0:02

score 180 · Answer 9 · 2026-03-25 00:04:25

180

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How to show in $R^2$, the $f(x,y)=(x^4+y^4)^{1/4}$ defines a norm on $R^2$.

Published on 25 Mar 2026 - 0:04

score 94 · Answer 10 · 2026-03-25 12:55:18

94

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How to prove the following floor function inequality?

Published on 25 Mar 2026 - 12:55

score 29 · Answer 11 · 2026-03-30 15:00:35

29

Views

Domain of Trigonometric Inequality

Published on 30 Mar 2026 - 15:00

score 183 · Answer 12 · 2026-03-26 02:35:45

183

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Prove that $\left(\sum_{k=1}^n\,\left(\frac{k}{n}\right)^n\right)^{\frac{1}{n}}$ decreases as $n$ increases.

Published on 26 Mar 2026 - 2:35

score 690 · Answer 13 · 2026-03-25 03:03:09

690

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show this inequality $ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$

Published on 25 Mar 2026 - 3:03

score 76 · Answer 14 · 2026-03-30 00:21:29

76

Views

Proof of the inequality of sums

Published on 30 Mar 2026 - 0:21

score 71 · Answer 15 · 2026-03-30 00:19:17

71

Views

Bounding the sum of powers

Published on 30 Mar 2026 - 0:19

List Question

Use the fact that $\ln(1+x) \leq x$ to show that $(a_1a_2\cdot\cdot\cdot a_n)^{1/n} \leq \frac{a_1+a_2+\cdot\cdot\cdot+a_n}{n}$

Find the minimize of $A=\frac{1}{a^2}\left(b^2+c^2\right)+a^2\left(\frac{1}{b^2}+\frac{1}{c^2}\right)$

Using the Mean Value Theorem, show that for all positive integers $n$, $n\ln(1+\frac{1}{n}) \leq 1$

Let $a,b,c$ be the length of the side $BC,CA,AB$ respectively for $\triangle ABC$. Show that $(2b+2c-2a)^3(a+b+c)\geq18a^2bc$.

Inequality from AMM problems section

Inequality in $n$-variables

What tools one should use for inequalities?

When does the equality hold in the Holder inequality?

How to show in $R^2$, the $f(x,y)=(x^4+y^4)^{1/4}$ defines a norm on $R^2$.

How to prove the following floor function inequality?

Domain of Trigonometric Inequality

Prove that $\left(\sum_{k=1}^n\,\left(\frac{k}{n}\right)^n\right)^{\frac{1}{n}}$ decreases as $n$ increases.

show this inequality $ab+bc+ac+\sin{(a-1)}+\sin{(b-1)}+\sin{(c-1)}\ge 3$

Proof of the inequality of sums

Bounding the sum of powers

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